Metamath Proof Explorer

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	$⊢ ((φ \to ψ) \to χ) \to (((ψ \to τ) \to (φ \to ⊥)) \to χ)$

Proof

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