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Theorem opelopabf 4722
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4719 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 4708 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
2 opelopabf.1 . . 3  |-  A  e. 
_V
3 nfcv 2616 . . . . 5  |-  F/_ x B
4 opelopabf.x . . . . 5  |-  F/ x ps
53, 4nfsbc 3316 . . . 4  |-  F/ x [. B  /  y ]. ps
6 opelopabf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
76sbcbidv 3353 . . . 4  |-  ( x  =  A  ->  ( [. B  /  y ]. ph  <->  [. B  /  y ]. ps ) )
85, 7sbciegf 3326 . . 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 3326 . . 3  |-  ( B  e.  _V  ->  ( [. B  /  y ]. ps  <->  ch ) )
1410, 13ax-mp 5 . 2  |-  ( [. B  /  y ]. ps  <->  ch )
151, 9, 143bitri 271 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370   F/wnf 1590    e. wcel 1758   _Vcvv 3078   [.wsbc 3294   <.cop 3992   {copab 4458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-opab 4460
This theorem is referenced by:  pofun  4766  fmptco  5986  fmptcof2  26131
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