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Metamath Proof Explorer

Theorem ax12vALT

Description: Alternate proof of ax12v2 , shorter, but depending on more axioms. (Contributed by NM, 5-Aug-1993) (New usage is discouraged.) (Proof modification is discouraged.)

		Ref	Expression
	Assertion	ax12vALT	$⊢ x = y \to (φ \to \forall x (x = y \to φ))$

Proof

Step	Hyp	Ref	Expression
1		ax-1	$⊢ φ \to (x = y \to φ)$
2		axc16	$⊢ \forall x x = y \to ((x = y \to φ) \to \forall x (x = y \to φ))$
3	1 2	syl5	$⊢ \forall x x = y \to (φ \to \forall x (x = y \to φ))$
4	3	a1d	$⊢ \forall x x = y \to (x = y \to (φ \to \forall x (x = y \to φ)))$
5		axc15	$⊢ \neg \forall x x = y \to (x = y \to (φ \to \forall x (x = y \to φ)))$
6	4 5	pm2.61i	$⊢ x = y \to (φ \to \forall x (x = y \to φ))$