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Theorem copsex2ga 6411
Description: Implicit substitution inference for ordered pairs. Compare copsex2g 4447. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
copsex2ga.1  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
copsex2ga  |-  ( A  e.  ( V  X.  W )  ->  ( ph 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ps ) ) )
Distinct variable groups:    x, y, A    ph, x, y
Allowed substitution hints:    ps( x, y)    V( x, y)    W( x, y)

Proof of Theorem copsex2ga
StepHypRef Expression
1 xpss 4985 . . 3  |-  ( V  X.  W )  C_  ( _V  X.  _V )
21sseli 3346 . 2  |-  ( A  e.  ( V  X.  W )  ->  A  e.  ( _V  X.  _V ) )
3 copsex2ga.1 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
43copsex2gb 6410 . . 3  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ps ) 
<->  ( A  e.  ( _V  X.  _V )  /\  ph ) )
54baibr 874 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( ph  <->  E. x E. y ( A  =  <. x ,  y >.  /\  ps ) ) )
62, 5syl 16 1  |-  ( A  e.  ( V  X.  W )  ->  ( ph 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819    X. cxp 4879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-opab 4270  df-xp 4887
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