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Theorem eqvinc 3619
 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 3618 . 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 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2115  Vcvv 3471 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2814  df-clel 2892 This theorem is referenced by:  eqvincf  3620  dff13  6987  f1eqcocnv  7031  tfindsg  7550  findsg  7584  findcard2s  8735  indpi  10306  fcoinvbr  30344  dfrdg4  33419  bj-elsngl  34296
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