Theorem nfeu1ALT 2588

Description: Alternate version of nfeu1 2589 with a shorter proof but using ax-12 2184. Bound-variable hypothesis builder for uniqueness. See also nfeu1 2589. (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 2574	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		nfexa2 2183	. 2 ⊢ Ⅎ𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
3	1, 2	nfxfr 1854	1 ⊢ Ⅎ𝑥∃!𝑥𝜑

Syntax hints:   ↔ wb 206  ∀wal 1539  ∃wex 1780  Ⅎwnf 1784  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-10 2146  ax-11 2162  ax-12 2184

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-nf 1785  df-mo 2539  df-eu 2569

This theorem is referenced by: (None)