Theorem adh-minimp-sylsimp

Description: Derivation of jarr (also called "syll-simp") from minimp and ax-mp . Polish prefix notation: CCCpqrCqr . (Contributed by BJ, 4-Apr-2021) (Revised by ADH, 10-Nov-2023) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	adh-minimp-sylsimp	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		adh-minimp-jarr-ax2c-lem3	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜒 ) )
2		adh-minimp-jarr-imim1-ax2c-lem1	⊢ ( ( ( 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 ) ) ) )
3		adh-minimp-jarr-lem2	⊢ ( ( ( ( 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 ) ) ) )
4	2 3	ax-mp	⊢ ( ( ( ( ( ( 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 ) ) )
5	1 4	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) )
6		adh-minimp-jarr-imim1-ax2c-lem1	⊢ ( ( ( ( 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 ) ) ) )
7		adh-minimp-jarr-lem2	⊢ ( ( ( ( ( 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 ) ) ) )
8	6 7	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) ) )
9	5 8	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → 𝜒 ) )