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Theorem eqrelriiv 4781
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 4780 . 2  |-  ( ( Rel  A  /\  Rel  B )  ->  A  =  B )
51, 2, 4mp2an 653 1  |-  A  =  B
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   <.cop 3643   Rel wrel 4694
This theorem is referenced by:  eqbrriv  4782  inopab  4816  difopab  4817  dfres2  5002  cnvopab  5083  cnv0  5084  cnvdif  5087  cnvcnvsn  5150  dfco2  5172  coiun  5182  co02  5186  coass  5191  ressn  5211  difxp  6153  ovoliunlem1  18861  h2hlm  21560  cnvco1  24117  cnvco2  24118  restidsing  25076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696
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