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Theorem opelopab 4605
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 16-May-1995.)
Hypotheses
Ref Expression
opelopab.1  |-  A  e. 
_V
opelopab.2  |-  B  e. 
_V
opelopab.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
opelopab.4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
opelopab  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
Distinct variable groups:    x, y, A    x, B, y    ch, x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem opelopab
StepHypRef Expression
1 opelopab.1 . 2  |-  A  e. 
_V
2 opelopab.2 . 2  |-  B  e. 
_V
3 opelopab.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
4 opelopab.4 . . 3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
53, 4opelopabg 4602 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ch ) )
61, 2, 5mp2an 672 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   _Vcvv 2967   <.cop 3878   {copab 4344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-opab 4346
This theorem is referenced by:  opabid2  4964  dfres2  5153  f1oiso  6037  elopabi  6630  xporderlem  6678  cnlnssadj  25435  areacirclem5  28441  pellexlem3  29125  dicopelval  34662  dih1dimatlem  34814
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