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Theorem opelopabf 4289
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4286 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 4275 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
2 opelopabf.1 . . 3  |-  A  e. 
_V
3 nfcv 2421 . . . . 5  |-  F/_ x B
4 opelopabf.x . . . . 5  |-  F/ x ps
53, 4nfsbc 3014 . . . 4  |-  F/ x [. B  /  y ]. ps
6 opelopabf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
76sbcbidv 3047 . . . 4  |-  ( x  =  A  ->  ( [. B  /  y ]. ph  <->  [. B  /  y ]. ps ) )
85, 7sbciegf 3024 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. ps ) )
92, 8ax-mp 10 . 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 3024 . . 3  |-  ( B  e.  _V  ->  ( [. B  /  y ]. ps  <->  ch ) )
1410, 13ax-mp 10 . 2  |-  ( [. B  /  y ]. ps  <->  ch )
151, 9, 143bitri 264 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   F/wnf 1532    = wceq 1624    e. wcel 1685   _Vcvv 2790   [.wsbc 2993   <.cop 3645   {copab 4078
This theorem is referenced by:  pofun  4330  fmptco  5653
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-opab 4080
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