Theorem eqvinc 3328

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 3327	. 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 196   ∧ wa 384   = wceq 1482  ∃wex 1703   ∈ wcel 1989  Vcvv 3198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-12 2046  ax-ext 2601

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1485  df-ex 1704  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3200

This theorem is referenced by:  eqvincf  3329  dff13  6509  f1eqcocnv  6553  tfindsg  7057  findsg  7090  findcard2s  8198  indpi  9726  fcoinvbr  29403  dfrdg4  32042  bj-elsngl  32940