Theorem nfeu1 2064

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 2056	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		nfa1 1563	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
3	2	nfex 1659	. 2 ⊢ Ⅎ𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
4	1, 3	nfxfr 1496	1 ⊢ Ⅎ𝑥∃!𝑥𝜑

Syntax hints:   ↔ wb 105  ∀wal 1370  Ⅎwnf 1482  ∃wex 1514  ∃!weu 2053

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-ial 1556

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-eu 2056

This theorem is referenced by:  nfmo1  2065  moaneu  2129  nfreu1  2677  eusv2i  4501  eusv2nf  4502  iota2  5260  sniota  5261  fv3  5598  tz6.12c  5605  eusvobj1  5930