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Metamath Proof Explorer

Theorem equs5aALT

Description: Alternate proof of equs5a . Uses ax-12 but not ax-13 . (Contributed by NM, 2-Feb-2007) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	equs5aALT	$⊢ \exists x (x = y \land \forall y φ) \to \forall x (x = y \to φ)$

Proof

Step	Hyp	Ref	Expression
1		nfa1	$⊢ Ⅎ x \forall x (x = y \to φ)$
2		ax-12	$⊢ x = y \to (\forall y φ \to \forall x (x = y \to φ))$
3	2	imp	$⊢ (x = y \land \forall y φ) \to \forall x (x = y \to φ)$
4	1 3	exlimi	$⊢ \exists x (x = y \land \forall y φ) \to \forall x (x = y \to φ)$