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Theorem eqrelrdvOLD 25876
Description: Deduce equality of relations from equivalence of membership. (Moved to eqrelrdv 4820 in main set.mm and may be deleted by mathbox owner, RM. --NM 20-Feb-2014.) (Contributed by Rodolfo Medina, 10-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
eqrelrdvOLD.1  |-  Rel  A
eqrelrdvOLD.2  |-  Rel  B
eqrelrdvOLD.3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
Assertion
Ref Expression
eqrelrdvOLD  |-  ( ph  ->  A  =  B )
Distinct variable groups:    x, y, A    x, B, y    ph, x, y

Proof of Theorem eqrelrdvOLD
StepHypRef Expression
1 eqrelrdvOLD.3 . . 3  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
21alrimivv 1623 . 2  |-  ( ph  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
3 eqrelrdvOLD.1 . . 3  |-  Rel  A
4 eqrelrdvOLD.2 . . 3  |-  Rel  B
5 eqrel 4814 . . 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 1531    = wceq 1633    e. wcel 1701   <.cop 3677   Rel wrel 4731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-opab 4115  df-xp 4732  df-rel 4733
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