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Theorem copsex2gb 6407
Description: Implicit substitution inference for ordered pairs. Compare copsex2ga 6408. (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 4936 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
21anbi1i 677 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  ph ) 
<->  ( E. x E. y  A  =  <. x ,  y >.  /\  ph ) )
3 19.41vv 1925 . 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 619 . . 3  |-  ( ( A  =  <. x ,  y >.  /\  ph ) 
<->  ( A  =  <. x ,  y >.  /\  ps ) )
652exbii 1593 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ps ) )
72, 3, 63bitr2ri 266 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 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817    X. cxp 4876
This theorem is referenced by:  copsex2ga  6408  elopaba  6409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884
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