Theorem ideq 4932

Description: For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)

Hypothesis

Ref	Expression
ideq.1	|- B e. _V

Assertion

Ref	Expression
ideq	|- ( A _I B <-> A = B )

Proof of Theorem ideq

Step	Hyp	Ref	Expression
1		ideq.1	. 2 |- B e. _V
2		ideqg 4931	. 2 |- ( B e. _V -> ( A _I B <-> A = B ) )
3	1, 2	ax-mp 5	1 |- ( A _I B <-> A = B )

This proof depends on syntax axioms:   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   _I cid 4433

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781

This theorem is used by:  dmi  4996  resieq  5073  resiexg  5108  iss  5109  restidsing  5119  imai  5143  issref  5170  intasym  5172  asymref  5173  intirr  5174  poirr2  5180  cnvi  5192  coi1  5303  idssen  7063