equs5eALT

Description: Alternate proof of equs5e . Uses ax-12 but not ax-13 . (Contributed by NM, 2-Feb-2007) (Proof shortened by Wolf Lammen, 15-Jan-2018) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	equs5eALT	$⊢ \exists x (x = y \land φ) \to \forall x (x = y \to \exists y φ)$

Proof

Step	Hyp	Ref	Expression
1		nfa1	$⊢ Ⅎ x \forall x (x = y \to \exists y φ)$
2		hbe1	$⊢ \exists y φ \to \forall y \exists y φ$
3	2	19.23bi	$⊢ φ \to \forall y \exists y φ$
4		ax-12	$⊢ x = y \to (\forall y \exists y φ \to \forall x (x = y \to \exists y φ))$
5	3 4	syl5	$⊢ x = y \to (φ \to \forall x (x = y \to \exists y φ))$
6	5	imp	$⊢ (x = y \land φ) \to \forall x (x = y \to \exists y φ)$
7	1 6	exlimi	$⊢ \exists x (x = y \land φ) \to \forall x (x = y \to \exists y φ)$

Metamath Proof Explorer

Theorem equs5eALT

Proof