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Theorem opelopaba 4282
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopaba.1  |-  A  e. 
_V
opelopaba.2  |-  B  e. 
_V
opelopaba.3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
opelopaba  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ps )
Distinct variable groups:    x, y, A   
x, B, y    ps, x, y
Allowed substitution groups:    ph( x, y)

Proof of Theorem opelopaba
StepHypRef Expression
1 opelopaba.1 . 2  |-  A  e. 
_V
2 opelopaba.2 . 2  |-  B  e. 
_V
3 opelopaba.3 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
43opelopabga 4279 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ps ) )
51, 2, 4mp2an 655 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1625    e. wcel 1687   _Vcvv 2791   <.cop 3646   {copab 4079
This theorem is referenced by:  canthwelem  8269  canthwe  8270  bcthlem1  18742
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pr 4215
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-rab 2555  df-v 2793  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-sn 3649  df-pr 3650  df-op 3652  df-opab 4081
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