Theorem 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	$$⊢\exists x\left(x=y\wedge \phi \right)\to \forall x\left(x=y\to \exists y\phi \right)$$

Proof

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