Metamath Proof Explorer

Theorem retbwax2

Description: tbw-ax2 rederived from merco1 . (Contributed by Anthony Hart, 17-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	retbwax2	$⊢ φ \to (ψ \to φ)$

Proof

Step	Hyp	Ref	Expression
1		merco1lem1	$⊢ ((((φ \to φ) \to φ) \to (φ \to ⊥)) \to φ) \to (⊥ \to φ)$
2		merco1	$⊢ (((((φ \to φ) \to φ) \to (φ \to ⊥)) \to φ) \to (⊥ \to φ)) \to (((⊥ \to φ) \to (φ \to φ)) \to (φ \to (φ \to φ)))$
3	1 2	ax-mp	$⊢ ((⊥ \to φ) \to (φ \to φ)) \to (φ \to (φ \to φ))$
4		merco1	$⊢ ((((φ \to (φ \to φ)) \to (φ \to ⊥)) \to (φ \to ⊥)) \to ⊥) \to ((⊥ \to φ) \to (φ \to φ))$
5		merco1	$⊢ (((((φ \to (φ \to φ)) \to (φ \to ⊥)) \to (φ \to ⊥)) \to ⊥) \to ((⊥ \to φ) \to (φ \to φ))) \to ((((⊥ \to φ) \to (φ \to φ)) \to (φ \to (φ \to φ))) \to (φ \to (φ \to (φ \to φ))))$
6	4 5	ax-mp	$⊢ (((⊥ \to φ) \to (φ \to φ)) \to (φ \to (φ \to φ))) \to (φ \to (φ \to (φ \to φ)))$
7	3 6	ax-mp	$⊢ φ \to (φ \to (φ \to φ))$
8		merco1lem1	$⊢ ((((ψ \to φ) \to φ) \to (φ \to ⊥)) \to φ) \to (⊥ \to φ)$
9		merco1	$⊢ (((((ψ \to φ) \to φ) \to (φ \to ⊥)) \to φ) \to (⊥ \to φ)) \to (((⊥ \to φ) \to (ψ \to φ)) \to (φ \to (ψ \to φ)))$
10	8 9	ax-mp	$⊢ ((⊥ \to φ) \to (ψ \to φ)) \to (φ \to (ψ \to φ))$
11		merco1	$⊢ ((((φ \to (ψ \to φ)) \to (ψ \to ⊥)) \to ((φ \to (φ \to (φ \to φ))) \to ⊥)) \to ⊥) \to ((⊥ \to φ) \to (ψ \to φ))$
12		merco1	$⊢ (((((φ \to (ψ \to φ)) \to (ψ \to ⊥)) \to ((φ \to (φ \to (φ \to φ))) \to ⊥)) \to ⊥) \to ((⊥ \to φ) \to (ψ \to φ))) \to ((((⊥ \to φ) \to (ψ \to φ)) \to (φ \to (ψ \to φ))) \to ((φ \to (φ \to (φ \to φ))) \to (φ \to (ψ \to φ))))$
13	11 12	ax-mp	$⊢ (((⊥ \to φ) \to (ψ \to φ)) \to (φ \to (ψ \to φ))) \to ((φ \to (φ \to (φ \to φ))) \to (φ \to (ψ \to φ)))$
14	10 13	ax-mp	$⊢ (φ \to (φ \to (φ \to φ))) \to (φ \to (ψ \to φ))$
15	7 14	ax-mp	$⊢ φ \to (ψ \to φ)$