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Theorem opelopab 4775
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 4771 . 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 1379    e. wcel 1767   _Vcvv 3118   <.cop 4039   {copab 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-opab 4512
This theorem is referenced by:  opabid2  5138  dfres2  5332  f1oiso  6246  elopabi  6856  xporderlem  6906  cnlnssadj  26830  areacirclem5  30045  pellexlem3  30701  dicopelval  36380  dih1dimatlem  36532
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