Theorem eqvinc 3637

Description: A variable introduction law for class equality. (Contributed by NM, 14-Apr-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Thierry Arnoux, 23-Jan-2022.)

Hypothesis

Ref	Expression
eqvinc.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
eqvinc	⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eqvinc

Step	Hyp	Ref	Expression
1		eqvinc.1	. 2 ⊢ 𝐴 ∈ V
2		eqvincg 3636	. 2 ⊢ (𝐴 ∈ V → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵)))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 = 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806

This theorem is referenced by:  eqvincf  3638  dff13  7271  f1eqcocnv  7316  f1eqcocnvOLD  7317  tfindsg  7871  findsg  7911  findcard2s  9196  indpi  10938  fcoinvbr  32416  dfrdg4  35580  bj-elsngl  36480