Metamath Proof Explorer


Theorem merco1lem6

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 . (Contributed by Anthony Hart, 17-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion merco1lem6 ( ( 𝜑 → ( 𝜑𝜓 ) ) → ( 𝜒 → ( 𝜑𝜓 ) ) )

Proof

Step Hyp Ref Expression
1 merco1lem5 ( ( ( ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) → ⊥ ) → ⊥ ) → ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) )
2 merco1lem3 ( ( ( ( ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) → ⊥ ) → ⊥ ) → ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) ) → ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ( ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) → ⊥ ) ) )
3 1 2 ax-mp ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ( ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) → ⊥ ) )
4 merco1lem5 ( ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ( ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) → ⊥ ) ) → ( ( 𝜑𝜓 ) → ( ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) → ⊥ ) ) )
5 3 4 ax-mp ( ( 𝜑𝜓 ) → ( ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) → ⊥ ) )
6 merco1lem3 ( ( ( 𝜑𝜓 ) → ( ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) → ⊥ ) ) → ( ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) → 𝜑 ) )
7 5 6 ax-mp ( ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) → 𝜑 )
8 merco1 ( ( ( ( ( ( 𝜑𝜓 ) → ⊥ ) → ( 𝜒 → ⊥ ) ) → ⊥ ) → 𝜑 ) → ( ( 𝜑 → ( 𝜑𝜓 ) ) → ( 𝜒 → ( 𝜑𝜓 ) ) ) )
9 7 8 ax-mp ( ( 𝜑 → ( 𝜑𝜓 ) ) → ( 𝜒 → ( 𝜑𝜓 ) ) )