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Theorem eqrelriv 4928
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 1670 . 2  |-  A. x A. y ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  B )
3 eqrel 4924 . 2  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
42, 3mpbiri 237 1  |-  ( ( Rel  A  /\  Rel  B )  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442    = wceq 1444    e. wcel 1887   <.cop 3974   Rel wrel 4839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-opab 4462  df-xp 4840  df-rel 4841
This theorem is referenced by:  eqrelriiv  4929  dfrel2  5286  coi1  5351  cnviin  5373
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