MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opelopaba Unicode version

Theorem opelopaba 4218
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 hints:    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 4215 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ps ) )
51, 2, 4mp2an 656 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2740   <.cop 3584   {copab 4016
This theorem is referenced by:  canthwelem  8205  canthwe  8206  bcthlem1  18673
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 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-rab 2523  df-v 2742  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-opab 4018
  Copyright terms: Public domain W3C validator