Theorem eqvinc 3571

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 3570	. 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 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817

This theorem is referenced by:  eqvincf  3572  dff13  7109  f1eqcocnv  7153  f1eqcocnvOLD  7154  tfindsg  7682  findsg  7720  findcard2s  8910  indpi  10594  fcoinvbr  30848  dfrdg4  34180  bj-elsngl  35085