Metamath Proof Explorer

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

Proof

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