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Theorem eqvinc 3480
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 3479 . 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 196  wa 382   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-tru 1634  df-ex 1853  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353
This theorem is referenced by:  eqvincf  3481  dff13  6655  f1eqcocnv  6699  tfindsg  7207  findsg  7240  findcard2s  8357  indpi  9931  fcoinvbr  29757  dfrdg4  32395  bj-elsngl  33287
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