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Theorem eqbrrdv 4913
Description: Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
eqbrrdv.1  |-  ( ph  ->  Rel  A )
eqbrrdv.2  |-  ( ph  ->  Rel  B )
eqbrrdv.3  |-  ( ph  ->  ( x A y  <-> 
x B y ) )
Assertion
Ref Expression
eqbrrdv  |-  ( ph  ->  A  =  B )
Distinct variable groups:    x, y, A    x, B, y    ph, x, y

Proof of Theorem eqbrrdv
StepHypRef Expression
1 eqbrrdv.3 . . . 4  |-  ( ph  ->  ( x A y  <-> 
x B y ) )
2 df-br 4154 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
3 df-br 4154 . . . 4  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
41, 2, 33bitr3g 279 . . 3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
54alrimivv 1639 . 2  |-  ( ph  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
6 eqbrrdv.1 . . 3  |-  ( ph  ->  Rel  A )
7 eqbrrdv.2 . . 3  |-  ( ph  ->  Rel  B )
8 eqrel 4905 . . 3  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
96, 7, 8syl2anc 643 . 2  |-  ( ph  ->  ( A  =  B  <->  A. x A. y (
<. x ,  y >.  e.  A  <->  <. x ,  y
>.  e.  B ) ) )
105, 9mpbird 224 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1717   <.cop 3760   class class class wbr 4153   Rel wrel 4823
This theorem is referenced by:  eqbrrdva  4982  oppcsect2  13927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-xp 4824  df-rel 4825
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