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Theorem opelopabaf 4478
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4476 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 4465 . 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 1629 . . . 4  |-  F/ x  B  e.  _V
7 opelopabaf.3 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
84, 5, 6, 7sbc2iegf 3227 . . 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 1553    = wceq 1652    e. wcel 1725   _Vcvv 2956   [.wsbc 3161   <.cop 3817   {copab 4265
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267
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