Theorem eqvinc 3330

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 e. _V

Assertion

Ref	Expression
eqvinc	|- ( A = B <-> E. 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 e. _V
2		eqvincg 3329	. 2 |- ( A e. _V -> ( A = B <-> E. x ( x = A /\ x = B ) ) )
3	1, 2	ax-mp 5	1 |- ( A = B <-> E. x ( x = A /\ x = B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  eqvincf  3331  dff13  6512  f1eqcocnv  6556  tfindsg  7060  findsg  7093  findcard2s  8201  indpi  9729  fcoinvbr  29419  dfrdg4  32058  bj-elsngl  32956