Theorem merco1lem3

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	merco1lem3	⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ( 𝜒 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		merco1lem2	⊢ ( ( ( 𝜑 → 𝜑 ) → ⊥ ) → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ⊥ ) )
2		retbwax2	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → 𝜑 ) ) ) )
3		merco1lem2	⊢ ( ( ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → 𝜑 ) ) → ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ⊥ ) → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ⊥ ) ) → ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → 𝜑 ) ) ) ) )
4	2 3	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ⊥ ) → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ⊥ ) ) → ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → 𝜑 ) ) ) )
5	1 4	ax-mp	⊢ ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → 𝜑 ) ) )
6		merco1lem2	⊢ ( ( ( 𝜒 → 𝜑 ) → ⊥ ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ⊥ ) )
7		retbwax2	⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ( 𝜒 → 𝜑 ) ) → ( ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → 𝜑 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ( 𝜒 → 𝜑 ) ) ) )
8		merco1lem2	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ( 𝜒 → 𝜑 ) ) → ( ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → 𝜑 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ( 𝜒 → 𝜑 ) ) ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ⊥ ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ⊥ ) ) → ( ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → 𝜑 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ( 𝜒 → 𝜑 ) ) ) ) )
9	7 8	ax-mp	⊢ ( ( ( ( 𝜒 → 𝜑 ) → ⊥ ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ⊥ ) ) → ( ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → 𝜑 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ( 𝜒 → 𝜑 ) ) ) )
10	6 9	ax-mp	⊢ ( ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → 𝜑 ) ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ( 𝜒 → 𝜑 ) ) )
11	5 10	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 → ⊥ ) ) → ( 𝜒 → 𝜑 ) )