merco1lem16

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	merco1lem16	$⊢ ((φ \to (ψ \to χ)) \to τ) \to ((φ \to χ) \to τ)$

Proof

Step	Hyp	Ref	Expression
1		merco1lem15	$⊢ (φ \to χ) \to (φ \to (ψ \to χ))$
2		merco1lem11	$⊢ ((φ \to χ) \to (φ \to (ψ \to χ))) \to ((((τ \to φ) \to ((φ \to χ) \to ⊥)) \to ⊥) \to (φ \to (ψ \to χ)))$
3	1 2	ax-mp	$⊢ (((τ \to φ) \to ((φ \to χ) \to ⊥)) \to ⊥) \to (φ \to (ψ \to χ))$
4		merco1	$⊢ ((((τ \to φ) \to ((φ \to χ) \to ⊥)) \to ⊥) \to (φ \to (ψ \to χ))) \to (((φ \to (ψ \to χ)) \to τ) \to ((φ \to χ) \to τ))$
5	3 4	ax-mp	$⊢ ((φ \to (ψ \to χ)) \to τ) \to ((φ \to χ) \to τ)$

Metamath Proof Explorer

Theorem merco1lem16

Proof