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Theorem copsex2ga 6340
Description: Implicit substitution inference for ordered pairs. Compare copsex2g 4378. (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 4915 . . 3  |-  ( V  X.  W )  C_  ( _V  X.  _V )
21sseli 3280 . 2  |-  ( A  e.  ( V  X.  W )  ->  A  e.  ( _V  X.  _V ) )
3 copsex2ga.1 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
43copsex2gb 6339 . . 3  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ps ) 
<->  ( A  e.  ( _V  X.  _V )  /\  ph ) )
54baibr 873 . 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 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2892   <.cop 3753    X. cxp 4809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-opab 4201  df-xp 4817
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