Theorem nfeu1ALT

Description: Alternate version of nfeu1 with a shorter proof but using ax-12 . Bound-variable hypothesis builder for uniqueness. See also nfeu1 . (Contributed by NM, 9-Jul-1994) (Revised by Mario Carneiro, 7-Oct-2016) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	nfeu1ALT	$$⊢Ⅎx\exists !x\phi$$

Proof

Step	Hyp	Ref	Expression
1		eu6	$$⊢\exists !x\phi \leftrightarrow \exists y\forall x\left(\phi \leftrightarrow x=y\right)$$
2		nfexa2	$$⊢Ⅎx\exists y\forall x\left(\phi \leftrightarrow x=y\right)$$
3	1 2	nfxfr	$$⊢Ⅎx\exists !x\phi$$