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	⊢ Ⅎ 𝑥 ∃! 𝑥 𝜑

Proof

Step	Hyp	Ref	Expression
1		eu6	⊢ ( ∃! 𝑥 𝜑 ↔ ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 ) )
2		nfexa2	⊢ Ⅎ 𝑥 ∃ 𝑦 ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝑦 )
3	1 2	nfxfr	⊢ Ⅎ 𝑥 ∃! 𝑥 𝜑