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Theorem opelopabf 4166
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4163 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by NM, 19-Dec-2008.)
Hypotheses
Ref Expression
opelopabf.x  |-  F/ x ps
opelopabf.y  |-  F/ y ch
opelopabf.1  |-  A  e. 
_V
opelopabf.2  |-  B  e. 
_V
opelopabf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
opelopabf.4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
opelopabf  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
Distinct variable groups:    x, y, A   
x, B, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    ch( x, y)

Proof of Theorem opelopabf
StepHypRef Expression
1 opelopabsb 4152 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
2 opelopabf.1 . . 3  |-  A  e. 
_V
3 nfcv 2258 . . . . 5  |-  F/_ x B
4 opelopabf.x . . . . 5  |-  F/ x ps
53, 4nfsbc 2902 . . . 4  |-  F/ x [. B  /  y ]. ps
6 opelopabf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
76sbcbidv 2939 . . . 4  |-  ( x  =  A  ->  ( [. B  /  y ]. ph  <->  [. B  /  y ]. ps ) )
85, 7sbciegf 2912 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. ps ) )
92, 8ax-mp 5 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  / 
y ]. ps )
10 opelopabf.2 . . 3  |-  B  e. 
_V
11 opelopabf.y . . . 4  |-  F/ y ch
12 opelopabf.4 . . . 4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
1311, 12sbciegf 2912 . . 3  |-  ( B  e.  _V  ->  ( [. B  /  y ]. ps  <->  ch ) )
1410, 13ax-mp 5 . 2  |-  ( [. B  /  y ]. ps  <->  ch )
151, 9, 143bitri 205 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1316   F/wnf 1421    e. wcel 1465   _Vcvv 2660   [.wsbc 2882   <.cop 3500   {copab 3958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960
This theorem is referenced by:  pofun  4204  fmptco  5554
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