Theorem ideq 3283

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 3282	. 2 |- (B e. V -> (AIB <-> A = B))
3	1, 2	ax-mp 7	1 |- (AIB <-> A = B)

Syntax hints:   <-> wb 146   = wceq 958   e. wcel 960  Vcvv 1814   class class class wbr 2624  Icid 2837

This theorem is referenced by:  ididg 3284  dmi 3332  resieq 3382  resiexg 3402  iss 3403  imai 3423  intasym 3444  asymref 3445  intirr 3447  cnvi 3453  coi1 3516  fcoi1 3651  fcoi2 3652  ider 4275  idssen 4412

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191

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