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Theorem copsex2gb 6180
Description: Implicit substitution inference for ordered pairs. Compare copsex2ga 6181. (Contributed by NM, 12-Mar-2014.)
Hypothesis
Ref Expression
copsex2ga.1  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
copsex2gb  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ps ) 
<->  ( A  e.  ( _V  X.  _V )  /\  ph ) )
Distinct variable groups:    x, y, A    ph, x, y
Allowed substitution hints:    ps( x, y)

Proof of Theorem copsex2gb
StepHypRef Expression
1 elvv 4748 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
21anbi1i 676 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  ph ) 
<->  ( E. x E. y  A  =  <. x ,  y >.  /\  ph ) )
3 19.41vv 1843 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ph ) 
<->  ( E. x E. y  A  =  <. x ,  y >.  /\  ph ) )
4 copsex2ga.1 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
54pm5.32i 618 . . 3  |-  ( ( A  =  <. x ,  y >.  /\  ph ) 
<->  ( A  =  <. x ,  y >.  /\  ps ) )
652exbii 1570 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ps ) )
72, 3, 63bitr2ri 265 1  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ps ) 
<->  ( A  e.  ( _V  X.  _V )  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687
This theorem is referenced by:  copsex2ga  6181  elopaba  6182
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695
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