Theorem nfeu1 2017

Description: Bound-variable hypothesis builder for uniqueness. (Contributed by NM, 9-Jul-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)

Assertion

Ref	Expression
nfeu1	⊢ Ⅎ𝑥∃!𝑥𝜑

Proof of Theorem nfeu1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eu 2009	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		nfa1 1521	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
3	2	nfex 1617	. 2 ⊢ Ⅎ𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
4	1, 3	nfxfr 1454	1 ⊢ Ⅎ𝑥∃!𝑥𝜑

Syntax hints:   ↔ wb 104  ∀wal 1333  Ⅎwnf 1440  ∃wex 1472  ∃!weu 2006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-eu 2009

This theorem is referenced by:  nfmo1  2018  moaneu  2082  nfreu1  2628  eusv2i  4414  eusv2nf  4415  iota2  5160  sniota  5161  fv3  5490  tz6.12c  5497  eusvobj1  5808