Theorem ideq 3267

Description: For sets, the identity relation is the same as equality.

Hypothesis

Ref	Expression
ideq.1	|- B e. V

Assertion

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

Proof of Theorem ideq

Step	Hyp	Ref	Expression
1		ideq.1	. 2 |- B e. V
2		ideqg 3266	. 2 |- (B e. V -> (AIB <-> A = B))
3	1, 2	ax-mp 7	1 |- (AIB <-> A = B)

Syntax hints:   <-> wb 146   = wceq 953   e. wcel 955  Vcvv 1802   class class class wbr 2609  Icid 2820

This theorem is referenced by:  ididg 3268  dmi 3315  resieq 3360  resiexg 3380  iss 3381  imai 3401  intasym 3422  asymref 3423  asymrefOLD 3425  intirr 3427  cnvi 3433  coi1 3496  fcoi1 3630  fcoi2 3631  ider 4253  idssen 4387

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175

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