Metamath Proof Explorer


Theorem merco1lem14

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 . (Contributed by Anthony Hart, 18-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merco1lem14 ( ( ( ( 𝜑𝜓 ) → 𝜓 ) → 𝜒 ) → ( 𝜑𝜒 ) )

Proof

Step Hyp Ref Expression
1 merco1lem13 ( ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) )
2 merco1lem8 ( ( ( ( ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) → 𝜑 ) → ( ( ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) → ⊥ ) ) → 𝜑 ) → ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) )
3 merco1 ( ( ( ( ( ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) → 𝜑 ) → ( ( ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) → ⊥ ) ) → 𝜑 ) → ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) → ( ( ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) → ( ( ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) ) )
4 2 3 ax-mp ( ( ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) → ( ( ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) )
5 merco1lem9 ( ( ( ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) → ( ( ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) ) → ( ( ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) )
6 4 5 ax-mp ( ( ( ( ( 𝜑𝜓 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) ) → ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) )
7 1 6 ax-mp ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) )
8 merco1lem12 ( ( 𝜑 → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( ( ( 𝜒𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) )
9 7 8 ax-mp ( ( ( ( 𝜒𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ( 𝜑𝜓 ) → 𝜓 ) )
10 merco1 ( ( ( ( ( 𝜒𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ( 𝜑𝜓 ) → 𝜓 ) ) → ( ( ( ( 𝜑𝜓 ) → 𝜓 ) → 𝜒 ) → ( 𝜑𝜒 ) ) )
11 9 10 ax-mp ( ( ( ( 𝜑𝜓 ) → 𝜓 ) → 𝜒 ) → ( 𝜑𝜒 ) )