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Theorem opelopabaf 4304
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4302 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 4291 . 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 1609 . . . 4  |-  F/ x  B  e.  _V
7 opelopabaf.3 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
84, 5, 6, 7sbc2iegf 3070 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  ps )
)
92, 3, 8mp2an 653 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  ps )
101, 9bitri 240 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1534    = wceq 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004   <.cop 3656   {copab 4092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094
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