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Theorem opelopabf 4471
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4468 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 4457 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
2 opelopabf.1 . . 3  |-  A  e. 
_V
3 nfcv 2571 . . . . 5  |-  F/_ x B
4 opelopabf.x . . . . 5  |-  F/ x ps
53, 4nfsbc 3174 . . . 4  |-  F/ x [. B  /  y ]. ps
6 opelopabf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
76sbcbidv 3207 . . . 4  |-  ( x  =  A  ->  ( [. B  /  y ]. ph  <->  [. B  /  y ]. ps ) )
85, 7sbciegf 3184 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. ps ) )
92, 8ax-mp 8 . 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 3184 . . 3  |-  ( B  e.  _V  ->  ( [. B  /  y ]. ps  <->  ch ) )
1410, 13ax-mp 8 . 2  |-  ( [. B  /  y ]. ps  <->  ch )
151, 9, 143bitri 263 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   F/wnf 1553    = wceq 1652    e. wcel 1725   _Vcvv 2948   [.wsbc 3153   <.cop 3809   {copab 4257
This theorem is referenced by:  pofun  4511  fmptco  5892  fmptcof2  24064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259
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