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Theorem opelopabaf 4419
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4417 uses bound-variable hypotheses in place of distinct variable conditions." (Contributed by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Hypotheses
Ref Expression
opelopabaf.x  |-  F/ x ps
opelopabaf.y  |-  F/ y ps
opelopabaf.1  |-  A  e. 
_V
opelopabaf.2  |-  B  e. 
_V
opelopabaf.3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
opelopabaf  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ps )
Distinct variable groups:    x, y, A    x, B, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem opelopabaf
StepHypRef Expression
1 opelopabsb 4406 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
2 opelopabaf.1 . . 3  |-  A  e. 
_V
3 opelopabaf.2 . . 3  |-  B  e. 
_V
4 opelopabaf.x . . . 4  |-  F/ x ps
5 opelopabaf.y . . . 4  |-  F/ y ps
6 nfv 1626 . . . 4  |-  F/ x  B  e.  _V
7 opelopabaf.3 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
84, 5, 6, 7sbc2iegf 3170 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  ps )
)
92, 3, 8mp2an 654 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  ps )
101, 9bitri 241 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   F/wnf 1550    = wceq 1649    e. wcel 1717   _Vcvv 2899   [.wsbc 3104   <.cop 3760   {copab 4206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-opab 4208
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