Theorem nfeu1 2097

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 2089	. 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2		nfa1 1594	. . 3 ⊢ Ⅎ𝑥∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
3	2	nfex 1690	. 2 ⊢ Ⅎ𝑥∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦)
4	1, 3	nfxfr 1527	1 ⊢ Ⅎ𝑥∃!𝑥𝜑

This proof depends on syntax axioms:   ↔ wb 105  ∀wal 1400  Ⅎwnf 1513  ∃wex 1545  ∃!weu 2086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-nf 1514  df-eu 2089

This theorem is used by:  nfmo1  2098  moaneu  2163  nfreu1  2723  eusv2i  4601  eusv2nf  4602  iota2  5367  sniota  5368  fv3  5718  tz6.12c  5725  eusvobj1  6072