Theorem nfeu1ALT 2616

Description: Alternate version of nfeu1 2617 with a shorter proof but using ax-12 2215. Bound-variable hypothesis builder for uniqueness. See also nfeu1 2617. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nfeu1ALT	⊢ Ⅎ𝑥∃!𝑥𝜑

Proof of Theorem nfeu1ALT

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eu6 2602	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		nfexa2 2214	. 2 ⊢ Ⅎ𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
3	1, 2	nfxfr 1885	1 ⊢ Ⅎ𝑥∃!𝑥𝜑

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1567  ∃wex 1811  Ⅎwnf 1815  ∃!weu 2596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-10 2179  ax-11 2196  ax-12 2215

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-nf 1816  df-mo 2567  df-eu 2597

This theorem is used by: (None)