Metamath Proof Explorer

Theorem minimp-syllsimp

Description: Derivation of Syll-Simp ( jarr ) from ax-mp and minimp . (Contributed by BJ, 4-Apr-2021) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion minimp-syllsimp ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) )

Proof

Step Hyp Ref Expression
1 minimp ⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) )
2 minimp ⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) ) ) )
3 minimp ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) ) )
4 2 3 ax-mp ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) )
5 minimp ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) ) ) )
6 minimp ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) )
7 minimp ⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) ) )
8 5 6 7 mp2 ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) )
9 1 4 8 mp2b ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) )
10 minimp ⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) )
11 minimp ⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) ) )
12 minimp ⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) ) ) )
13 minimp ⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) )
14 minimp ⊢ ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) ) ) )
15 12 13 14 mp2 ⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) )
16 10 11 15 mp2b ⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) )
17 minimp ⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) ) )
18 minimp ⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) → ( ( ( 𝜓 → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) ) )
19 minimp ⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) ) ) )
20 minimp ⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( 𝜓 → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( 𝜓 → 𝜒 ) ) ) )
21 minimp ⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( 𝜓 → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) → ( ( ( 𝜓 → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) ) ) )
22 19 20 21 mp2 ⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) ) → ( ( ( 𝜓 → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) )
23 17 18 22 mp2b ⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) )
24 9 16 23 mp2b ⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) )