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Theorem eqrelrdv2 4802
Description: A version of eqrelrdv 4799. (Contributed by Rodolfo Medina, 10-Oct-2010.)
Hypothesis
Ref Expression
eqrelrdv2.1  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
Assertion
Ref Expression
eqrelrdv2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Distinct variable groups:    x, y, A    x, B, y    ph, x, y

Proof of Theorem eqrelrdv2
StepHypRef Expression
1 eqrelrdv2.1 . . . 4  |-  ( ph  ->  ( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
21alrimivv 1622 . . 3  |-  ( ph  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
32adantl 452 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
4 eqrel 4793 . . 3  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
54adantr 451 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
63, 5mpbird 223 1  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   <.cop 3656   Rel wrel 4710
This theorem is referenced by:  xpiindi  4837  fliftcnv  5826  dmtpos  6262  ercnv  6697  fpwwe2lem9  8276  invsym2  13681  eqbrrdv2  26834  dibglbN  31978  diclspsn  32006  dih1  32098  dihglbcpreN  32112  dihmeetlem4preN  32118
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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