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Theorem copsex2gb 6222
Description: Implicit substitution inference for ordered pairs. Compare copsex2ga 6223. (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 4785 . . 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 1874 . 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 1574 . 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 1532    = wceq 1633    e. wcel 1701   _Vcvv 2822   <.cop 3677    X. cxp 4724
This theorem is referenced by:  copsex2ga  6223  elopaba  6224
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
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