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Theorem eqrelriv 4780
Description: Inference from extensionality principle for relations. (Contributed by FL, 15-Oct-2012.)
Hypothesis
Ref Expression
eqrelriv.1  |-  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )
Assertion
Ref Expression
eqrelriv  |-  ( ( Rel  A  /\  Rel  B )  ->  A  =  B )
Distinct variable groups:    x, y, A    x, B, y

Proof of Theorem eqrelriv
StepHypRef Expression
1 eqrelriv.1 . . 3  |-  ( <.
x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B )
21gen2 1534 . 2  |-  A. x A. y ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  B )
3 eqrel 4777 . 2  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
42, 3mpbiri 224 1  |-  ( ( Rel  A  /\  Rel  B )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   <.cop 3643   Rel wrel 4694
This theorem is referenced by:  eqrelriiv  4781  dfrel2  5124  coi1  5188  cnviin  5212
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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