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Theorem copsex2ga 6197
Description: Implicit substitution inference for ordered pairs. Compare copsex2g 4270. (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 4809 . . 3  |-  ( V  X.  W )  C_  ( _V  X.  _V )
21sseli 3189 . 2  |-  ( A  e.  ( V  X.  W )  ->  A  e.  ( _V  X.  _V ) )
3 copsex2ga.1 . . . 4  |-  ( A  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
43copsex2gb 6196 . . 3  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ps ) 
<->  ( A  e.  ( _V  X.  _V )  /\  ph ) )
54baibr 872 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  ( ph  <->  E. x E. y ( A  =  <. x ,  y >.  /\  ps ) ) )
62, 5syl 15 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 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    X. cxp 4703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711
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