Theorem merco1lem14

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	merco1lem14	⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		merco1lem13	⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) )
2		merco1lem8	⊢ ( ( ( ( ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → 𝜑 ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) → ⊥ ) ) → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) )
3		merco1	⊢ ( ( ( ( ( ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → 𝜑 ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) → ⊥ ) ) → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) ) )
4	2 3	ax-mp	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) )
5		merco1lem9	⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) )
6	4 5	ax-mp	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) → ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) )
7	1 6	ax-mp	⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
8		merco1lem12	⊢ ( ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) )
9	7 8	ax-mp	⊢ ( ( ( ( 𝜒 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) )
10		merco1	⊢ ( ( ( ( ( 𝜒 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )
11	9 10	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) )