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Theorem eqrelrdv 4799
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 1622 . 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 4793 . . 3  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
63, 4, 5mp2an 653 . 2  |-  ( A  =  B  <->  A. x A. y ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  B ) )
72, 6sylibr 203 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    = wceq 1632    e. wcel 1696   <.cop 3656   Rel wrel 4710
This theorem is referenced by:  eqbrrdiv  4801  fcnvres  5434  fmptco  5707  fpwwe2lem8  8275  fpwwe2lem12  8279  fsumcom2  12253  gsumcom2  15242  lgsquadlem1  20609  lgsquadlem2  20610  fmptcof2  23244  dih1dimatlem  32141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-rel 4712
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