Metamath Proof Explorer


Theorem merco1lem16

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 merco1lem16 ( ( ( 𝜑 → ( 𝜓𝜒 ) ) → 𝜏 ) → ( ( 𝜑𝜒 ) → 𝜏 ) )

Proof

Step Hyp Ref Expression
1 merco1lem15 ( ( 𝜑𝜒 ) → ( 𝜑 → ( 𝜓𝜒 ) ) )
2 merco1lem11 ( ( ( 𝜑𝜒 ) → ( 𝜑 → ( 𝜓𝜒 ) ) ) → ( ( ( ( 𝜏𝜑 ) → ( ( 𝜑𝜒 ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → ( 𝜓𝜒 ) ) ) )
3 1 2 ax-mp ( ( ( ( 𝜏𝜑 ) → ( ( 𝜑𝜒 ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → ( 𝜓𝜒 ) ) )
4 merco1 ( ( ( ( ( 𝜏𝜑 ) → ( ( 𝜑𝜒 ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → ( 𝜓𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜓𝜒 ) ) → 𝜏 ) → ( ( 𝜑𝜒 ) → 𝜏 ) ) )
5 3 4 ax-mp ( ( ( 𝜑 → ( 𝜓𝜒 ) ) → 𝜏 ) → ( ( 𝜑𝜒 ) → 𝜏 ) )