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

Proof

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