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Theorem copsex2gb 4964
Description: Implicit substitution inference for ordered pairs. Compare copsex2ga 4965. (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 4912 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
21anbi1i 706 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  ph ) 
<->  ( E. x E. y  A  =  <. x ,  y >.  /\  ph ) )
3 19.41vv 1842 . 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 647 . . 3  |-  ( ( A  =  <. x ,  y >.  /\  ph ) 
<->  ( A  =  <. x ,  y >.  /\  ps ) )
652exbii 1730 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ph ) 
<->  E. x E. y
( A  =  <. x ,  y >.  /\  ps ) )
72, 3, 63bitr2ri 282 1  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  ps ) 
<->  ( A  e.  ( _V  X.  _V )  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455   E.wex 1674    e. wcel 1898   _Vcvv 3057   <.cop 3986    X. cxp 4851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-opab 4476  df-xp 4859
This theorem is referenced by:  copsex2ga  4965  elopaba  4966  elcnvlem  36252
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