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Theorem eqrelriv 4916
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 1640 . 2  |-  A. x A. y ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  B )
3 eqrel 4912 . 2  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
42, 3mpbiri 233 1  |-  ( ( Rel  A  /\  Rel  B )  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1403    = wceq 1405    e. wcel 1842   <.cop 3977   Rel wrel 4827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-opab 4453  df-xp 4828  df-rel 4829
This theorem is referenced by:  eqrelriiv  4917  dfrel2  5273  coi1  5338  cnviin  5360
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