Theorem mercolem7

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	mercolem7	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) )

Proof

Step	Hyp	Ref	Expression
1		merco2	⊢ ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) )
2		mercolem3	⊢ ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) ) )
3		mercolem6	⊢ ( ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) ) ) → ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) ) )
4	2 3	ax-mp	⊢ ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) )
5		mercolem5	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) ) )
6		mercolem4	⊢ ( ( 𝜑 → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) ) ) → ( ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) ) ) ) )
7	5 6	ax-mp	⊢ ( ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) ) ) )
8	4 7	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) ) )
9	1 8	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜑 → 𝜒 ) → ( 𝜃 → 𝜓 ) ) → ( 𝜃 → 𝜓 ) ) )