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Theorem eqrelrdv 5090
Description: Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.)
Hypotheses
Ref Expression
eqrelrdv.1  |-  Rel  A
eqrelrdv.2  |-  Rel  B
eqrelrdv.3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
Assertion
Ref Expression
eqrelrdv  |-  ( ph  ->  A  =  B )
Distinct variable groups:    x, y, A    x, B, y    ph, x, y

Proof of Theorem eqrelrdv
StepHypRef Expression
1 eqrelrdv.3 . . 3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
21alrimivv 1691 . 2  |-  ( ph  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
3 eqrelrdv.1 . . 3  |-  Rel  A
4 eqrelrdv.2 . . 3  |-  Rel  B
5 eqrel 5083 . . 3  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
63, 4, 5mp2an 672 . 2  |-  ( A  =  B  <->  A. x A. y ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  B ) )
72, 6sylibr 212 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1372    = wceq 1374    e. wcel 1762   <.cop 4026   Rel wrel 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-opab 4499  df-xp 4998  df-rel 4999
This theorem is referenced by:  eqbrrdiv  5092  fcnvres  5753  fmptco  6045  fpwwe2lem8  9004  fpwwe2lem12  9008  fsumcom2  13538  gsumcom2  16787  lgsquadlem1  23350  lgsquadlem2  23351  fmptcof2  27160  dfcnv2  27175  fprodcom2  28677  dih1dimatlem  36001
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