Theorem merco1lem12

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	merco1lem12	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		merco1lem3	⊢ ( ( ( ( 𝜑 → 𝜏 ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → ⊥ ) ) → ( 𝜒 → ⊥ ) ) → ( 𝜒 → ( 𝜑 → 𝜏 ) ) )
2		merco1	⊢ ( ( ( ( ( 𝜑 → 𝜏 ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → ⊥ ) ) → ( 𝜒 → ⊥ ) ) → ( 𝜒 → ( 𝜑 → 𝜏 ) ) ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → 𝜑 ) ) )
3	1 2	ax-mp	⊢ ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → 𝜑 ) )
4		merco1lem9	⊢ ( ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → 𝜑 ) ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → 𝜑 ) )
5	3 4	ax-mp	⊢ ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → 𝜑 )
6		merco1lem11	⊢ ( ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜑 ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) )
7	5 6	ax-mp	⊢ ( ( ( ( 𝜓 → 𝜑 ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 )
8		merco1	⊢ ( ( ( ( ( 𝜓 → 𝜑 ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → 𝜓 ) ) )
9	7 8	ax-mp	⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜒 → ( 𝜑 → 𝜏 ) ) → 𝜑 ) → 𝜓 ) )