Theorem merco1lem2

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	merco1lem2	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( ( 𝜓 → 𝜏 ) → ( 𝜑 → ⊥ ) ) → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		retbwax2	⊢ ( ( ( ( 𝜓 → 𝜏 ) → ( 𝜑 → ⊥ ) ) → ⊥ ) → ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜓 → 𝜏 ) → ( 𝜑 → ⊥ ) ) → ⊥ ) ) )
2		merco1	⊢ ( ( ( ( ( 𝜓 → 𝜏 ) → ( 𝜑 → ⊥ ) ) → ⊥ ) → ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜓 → 𝜏 ) → ( 𝜑 → ⊥ ) ) → ⊥ ) ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜓 → 𝜏 ) → ( 𝜑 → ⊥ ) ) → ⊥ ) ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) )
3	1 2	ax-mp	⊢ ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜓 → 𝜏 ) → ( 𝜑 → ⊥ ) ) → ⊥ ) ) → 𝜓 ) → ( 𝜑 → 𝜓 ) )
4		merco1	⊢ ( ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜓 → 𝜏 ) → ( 𝜑 → ⊥ ) ) → ⊥ ) ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( ( 𝜓 → 𝜏 ) → ( 𝜑 → ⊥ ) ) → 𝜒 ) ) )
5	3 4	ax-mp	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜒 ) → ( ( ( 𝜓 → 𝜏 ) → ( 𝜑 → ⊥ ) ) → 𝜒 ) )