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Theorem opelopabga 4081
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypothesis
Ref Expression
opelopabga.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
opelopabga  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. 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)    V( x, y)    W( x, y)

Proof of Theorem opelopabga
StepHypRef Expression
1 elopab 4076 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) )
2 opelopabga.1 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
32copsex2g 4064 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. x E. y ( <. A ,  B >.  =  <. x ,  y >.  /\  ph ) 
<->  ps ) )
41, 3syl5bb 190 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   <.cop 3444   {copab 3890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892
This theorem is referenced by:  brabga  4082  opelopab2a  4083  opelopaba  4084  opelopabg  4086  isprmpt2  5990  hashennn  10153
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