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Theorem eqvinc 3639
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 = 𝐴𝑥 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem eqvinc
StepHypRef Expression
1 eqvinc.1 . 2 𝐴 ∈ V
2 eqvincg 3638 . 2 (𝐴 ∈ V → (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵)))
31, 2ax-mp 5 1 (𝐴 = 𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2811  df-clel 2890
This theorem is referenced by:  eqvincf  3640  dff13  7004  f1eqcocnv  7048  tfindsg  7564  findsg  7598  findcard2s  8747  indpi  10317  fcoinvbr  30286  dfrdg4  33309  bj-elsngl  34177
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