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Theorem eqrelriiv 4971
Description: Inference from extensionality principle for relations. (Contributed by NM, 17-Mar-1995.)
Hypotheses
Ref Expression
eqreliiv.1  |-  Rel  A
eqreliiv.2  |-  Rel  B
eqreliiv.3  |-  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )
Assertion
Ref Expression
eqrelriiv  |-  A  =  B
Distinct variable groups:    x, y, A    x, B, y

Proof of Theorem eqrelriiv
StepHypRef Expression
1 eqreliiv.1 . 2  |-  Rel  A
2 eqreliiv.2 . 2  |-  Rel  B
3 eqreliiv.3 . . 3  |-  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )
43eqrelriv 4970 . 2  |-  ( ( Rel  A  /\  Rel  B )  ->  A  =  B )
51, 2, 4mp2an 655 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   <.cop 3818   Rel wrel 4884
This theorem is referenced by:  eqbrriv  4972  inopab  5006  difopab  5007  dfres2  5194  cnvopab  5275  cnv0  5276  cnvdif  5279  cnvcnvsn  5348  dfco2  5370  coiun  5380  co02  5384  coass  5389  ressn  5409  difxp  6381  ovoliunlem1  19399  h2hlm  22484  cnvco1  25384  cnvco2  25385
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 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
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 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-opab 4268  df-xp 4885  df-rel 4886
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