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Theorem opelopabaf 4181
Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. This version of opelopab 4179 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 4168 . 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 2987 . . 3  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( [. A  /  x ]. [. B  / 
y ]. ph  <->  ps )
)
92, 3, 8mp2an 656 . 2  |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  ps )
101, 9bitri 242 1  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ph }  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   F/wnf 1539    = wceq 1619    e. wcel 1621   _Vcvv 2727   [.wsbc 2921   <.cop 3547   {copab 3973
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-opab 3975
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