Theorem nfeu1ALT 2622

Description: Alternate version of nfeu1 2623 with a shorter proof but using ax-12 2219. Bound-variable hypothesis builder for uniqueness. See also nfeu1 2623. (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 2608	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		nfexa2 2218	. 2 ⊢ Ⅎ𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
3	1, 2	nfxfr 1880	1 ⊢ Ⅎ𝑥∃!𝑥𝜑

Syntax hints:   ↔ wb 209  ∀wal 1565  ∃wex 1806  Ⅎwnf 1810  ∃!weu 2602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-10 2182  ax-11 2198  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1807  df-nf 1811  df-mo 2573  df-eu 2603

This theorem is referenced by: (None)