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Theorem ideq 4870
 Description: For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.) (Revised by set.mm contributors, 1-Jun-2008.)
Hypothesis
Ref Expression
ideq.1 B V
Assertion
Ref Expression
ideq (A I BA = B)

Proof of Theorem ideq
StepHypRef Expression
1 ideq.1 . 2 B V
2 ideqg 4868 . 2 (B V → (A I BA = B))
31, 2ax-mp 5 1 (A I BA = B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2859   class class class wbr 4639   I cid 4763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-id 4767 This theorem is referenced by:  dmi  4919  iss  5000  imai  5010  intasym  5028  intirr  5029  cnvi  5032  coi1  5094  dffun2  5119  fvi  5442  dfid4  5503  elfix  5787  composeex  5820  funsex  5828  pw1fnex  5852  fnfullfunlem1  5856  ranfnex  5871  antisymex  5912  connexex  5913  foundex  5914  extex  5915  symex  5916  ider  5943  idssen  6040  tcfnex  6244  fnfreclem1  6317
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