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Theorem resieq 4837
 Description: A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
resieq

Proof of Theorem resieq
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 breq2 3941 . . . . 5
2 eqeq2 2150 . . . . 5
31, 2bibi12d 234 . . . 4
43imbi2d 229 . . 3
5 vex 2692 . . . . 5
65opres 4836 . . . 4
7 df-br 3938 . . . 4
85ideq 4699 . . . . 5
9 df-br 3938 . . . . 5
108, 9bitr3i 185 . . . 4
116, 7, 103bitr4g 222 . . 3
124, 11vtoclg 2749 . 2
1312impcom 124 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1332   wcel 1481  cop 3535   class class class wbr 3937   cid 4218   cres 4549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-res 4559 This theorem is referenced by:  foeqcnvco  5699  f1eqcocnv  5700
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