Theorem nfeu1 2053

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 2045	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		nfa1 1552	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
3	2	nfex 1648	. 2 ⊢ Ⅎ𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
4	1, 3	nfxfr 1485	1 ⊢ Ⅎ𝑥∃!𝑥𝜑

Syntax hints:   ↔ wb 105  ∀wal 1362  Ⅎwnf 1471  ∃wex 1503  ∃!weu 2042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-eu 2045

This theorem is referenced by:  nfmo1  2054  moaneu  2118  nfreu1  2666  eusv2i  4487  eusv2nf  4488  iota2  5245  sniota  5246  fv3  5578  tz6.12c  5585  eusvobj1  5906