Metamath Proof Explorer


Theorem merco1lem10

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 merco1lem10 ( ( ( ( ( 𝜑𝜓 ) → 𝜒 ) → ( 𝜏𝜒 ) ) → 𝜑 ) → ( 𝜃𝜑 ) )

Proof

Step Hyp Ref Expression
1 merco1 ( ( ( ( ( 𝜒𝜑 ) → ( 𝜏 → ⊥ ) ) → 𝜑 ) → ( 𝜑𝜓 ) ) → ( ( ( 𝜑𝜓 ) → 𝜒 ) → ( 𝜏𝜒 ) ) )
2 merco1lem2 ( ( ( ( ( ( 𝜒𝜑 ) → ( 𝜏 → ⊥ ) ) → 𝜑 ) → ( 𝜑𝜓 ) ) → ( ( ( 𝜑𝜓 ) → 𝜒 ) → ( 𝜏𝜒 ) ) ) → ( ( ( ( 𝜑𝜓 ) → ( 𝜃 → ⊥ ) ) → ( ( ( ( 𝜒𝜑 ) → ( 𝜏 → ⊥ ) ) → 𝜑 ) → ⊥ ) ) → ( ( ( 𝜑𝜓 ) → 𝜒 ) → ( 𝜏𝜒 ) ) ) )
3 1 2 ax-mp ( ( ( ( 𝜑𝜓 ) → ( 𝜃 → ⊥ ) ) → ( ( ( ( 𝜒𝜑 ) → ( 𝜏 → ⊥ ) ) → 𝜑 ) → ⊥ ) ) → ( ( ( 𝜑𝜓 ) → 𝜒 ) → ( 𝜏𝜒 ) ) )
4 merco1 ( ( ( ( ( 𝜑𝜓 ) → ( 𝜃 → ⊥ ) ) → ( ( ( ( 𝜒𝜑 ) → ( 𝜏 → ⊥ ) ) → 𝜑 ) → ⊥ ) ) → ( ( ( 𝜑𝜓 ) → 𝜒 ) → ( 𝜏𝜒 ) ) ) → ( ( ( ( ( 𝜑𝜓 ) → 𝜒 ) → ( 𝜏𝜒 ) ) → 𝜑 ) → ( 𝜃𝜑 ) ) )
5 3 4 ax-mp ( ( ( ( ( 𝜑𝜓 ) → 𝜒 ) → ( 𝜏𝜒 ) ) → 𝜑 ) → ( 𝜃𝜑 ) )