Theorem mercolem1

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 . (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	mercolem1	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		merco2	⊢ ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) )
2		merco2	⊢ ( ( ( 𝜒 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) )
3		merco2	⊢ ( ( ( 𝜓 → ( 𝜃 → 𝜒 ) ) → ( ( ⊥ → 𝜑 ) → ⊥ ) ) → ( ( ⊥ → 𝜓 ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) )
4		merco2	⊢ ( ( ( ( 𝜓 → ( 𝜃 → 𝜒 ) ) → ( ( ⊥ → 𝜑 ) → ⊥ ) ) → ( ( ⊥ → 𝜓 ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) ) → ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) → ( ( ⊥ → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) ) ) )
5	3 4	ax-mp	⊢ ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) → ( ( ⊥ → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) ) )
6		merco2	⊢ ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) → ( ( ⊥ → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ⊥ → 𝜑 ) → ( ( 𝜒 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) ) ) )
7	5 6	ax-mp	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ⊥ → 𝜑 ) → ( ( 𝜒 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) ) )
8		merco2	⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ⊥ → 𝜑 ) → ( ( 𝜒 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) ) ) ) )
9	7 8	ax-mp	⊢ ( ( ( ( 𝜒 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) ) ) )
10	2 9	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) ) )
11	1 10	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) ) )
12	1 11	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( 𝜓 → ( 𝜃 → 𝜒 ) ) )