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Theorem eqrelrdv 4974
 Description: Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)
Hypotheses
Ref Expression
eqrelrdv.1
eqrelrdv.2
eqrelrdv.3
Assertion
Ref Expression
eqrelrdv
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem eqrelrdv
StepHypRef Expression
1 eqrelrdv.3 . . 3
21alrimivv 1643 . 2
3 eqrelrdv.1 . . 3
4 eqrelrdv.2 . . 3
5 eqrel 4967 . . 3
63, 4, 5mp2an 655 . 2
72, 6sylibr 205 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178  wal 1550   wceq 1653   wcel 1726  cop 3819   wrel 4885 This theorem is referenced by:  eqbrrdiv  4976  fcnvres  5622  fmptco  5903  fpwwe2lem8  8514  fpwwe2lem12  8518  fsumcom2  12560  gsumcom2  15551  lgsquadlem1  21140  lgsquadlem2  21141  fmptcof2  24078  fprodcom2  25310  dih1dimatlem  32129 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4269  df-xp 4886  df-rel 4887
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