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Theorem eqbrrdv2 32403
Description: Other version of eqbrrdiv 4952. (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 4424 . . . 4  |-  ( x A y  <->  <. x ,  y >.  e.  A
)
3 df-br 4424 . . . 4  |-  ( x B y  <->  <. x ,  y >.  e.  B
)
41, 2, 33bitr3g 290 . . 3  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  B ) )
54eqrelrdv2 4953 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  (
( Rel  A  /\  Rel  B )  /\  ph ) )  ->  A  =  B )
65anabss5 823 1  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   <.cop 4004   class class class wbr 4423   Rel wrel 4858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-xp 4859  df-rel 4860
This theorem is referenced by:  prter3  32422
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