Theorem ideq 4880

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 4879	. 2 |- ( B e. _V -> ( A _I B <-> A = B ) )
3	1, 2	ax-mp 5	1 |- ( A _I B <-> A = B )

Syntax hints:   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2800   class class class wbr 4086   _I cid 4383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730

This theorem is referenced by:  dmi  4944  resieq  5021  resiexg  5056  iss  5057  restidsing  5067  imai  5090  issref  5117  intasym  5119  asymref  5120  intirr  5121  poirr2  5127  cnvi  5139  coi1  5250  idssen  6945