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Theorem eqrelrdv2OLD 26052
Description: Another version of eqrelrdv 4862. (Moved to eqrelrdv2 4865 in main set.mm and may be deleted by mathbox owner, RM. --NM 20-Feb-2014.) (Contributed by Rodolfo Medina, 30-Sep-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
eqrelrdv2OLD.1  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  B ) )
Assertion
Ref Expression
eqrelrdv2OLD  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A  =  B )
Distinct variable groups:    x, y, A    x, B, y    ph, x, y

Proof of Theorem eqrelrdv2OLD
StepHypRef Expression
1 eqrelrdv2OLD.1 . . 3  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( <. x ,  y >.  e.  A  <->  <.
x ,  y >.  e.  B ) )
21alrimivv 1632 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) )
3 eqrel 4856 . . 3  |-  ( ( Rel  A  /\  Rel  B )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
43adantr 451 . 2  |-  ( ( ( Rel  A  /\  Rel  B )  /\  ph )  ->  ( A  =  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  <->  <. x ,  y >.  e.  B
) ) )
52, 4mpbird 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 1540    = wceq 1642    e. wcel 1710   <.cop 3719   Rel wrel 4773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-opab 4157  df-xp 4774  df-rel 4775
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