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Theorem eqbrrdv2 26714
Description: Other version of eqbrrdiv 4976. (Contributed by Rodolfo Medina, 30-Sep-2010.)
Hypothesis
Ref Expression
eqbrrdv2.1  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( x A y  <->  x B y ) )
Assertion
Ref Expression
eqbrrdv2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Distinct variable groups:    x, y, A    x, B, y    ph, x, y

Proof of Theorem eqbrrdv2
StepHypRef Expression
1 eqbrrdv2.1 . . . 4  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( x A y  <->  x B y ) )
2 df-br 4215 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
3 df-br 4215 . . . 4  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
41, 2, 33bitr3g 280 . . 3  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  B ) )
54eqrelrdv2 4977 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  (
( Rel  A  /\  Rel  B )  /\  ph ) )  ->  A  =  B )
65anabss5 791 1  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   <.cop 3819   class class class wbr 4214   Rel wrel 4885
This theorem is referenced by:  prter3  26733
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887
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