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Theorem opelopabf 4226
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4223 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 4212 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [. A  /  x ]. [. B  / 
y ]. ph )
2 opelopabf.1 . . 3  |-  A  e. 
_V
3 nfcv 2392 . . . . 5  |-  F/_ x B
4 opelopabf.x . . . . 5  |-  F/ x ps
53, 4nfsbc 2956 . . . 4  |-  F/ x [. B  /  y ]. ps
6 opelopabf.3 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
76sbcbidv 2989 . . . 4  |-  ( x  =  A  ->  ( [. B  /  y ]. ph  <->  [. B  /  y ]. ps ) )
85, 7sbciegf 2966 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. B  /  y ]. ps ) )
92, 8ax-mp 10 . 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 2966 . . 3  |-  ( B  e.  _V  ->  ( [. B  /  y ]. ps  <->  ch ) )
1410, 13ax-mp 10 . 2  |-  ( [. B  /  y ]. ps  <->  ch )
151, 9, 143bitri 264 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ch )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   F/wnf 1539    = wceq 1619    e. wcel 1621   _Vcvv 2740   [.wsbc 2935   <.cop 3584   {copab 4016
This theorem is referenced by:  pofun  4267  fmptco  5590
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-opab 4018
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