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Theorem opelopabg 4255
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopabg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
opelopabg.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
opelopabg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. 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)    V( x, y)    W( x, y)

Proof of Theorem opelopabg
StepHypRef Expression
1 opelopabg.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2 opelopabg.2 . . 3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
31, 2sylan9bb 683 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ch )
)
43opelopabga 4250 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   <.cop 3617   {copab 4050
This theorem is referenced by:  opelopab  4258  fvopab3g  5532  fvopab3ig  5533  ov  5901  ovg  5920  eltopspOLD  16618  istpsOLD  16620  iscom2  21039  isdivrngo  21058  isvclem  21093  adj1  22473  adjeq  22475  linedegen  24141  islatalg  24550  cmppar3  25341  opelopab3  25740  dihpN  30693
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-opab 4052
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