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Theorem bnj1468 28548
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1468.1  |-  ( ps 
->  A. x ps )
bnj1468.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
bnj1468.3  |-  ( y  e.  A  ->  A. x  y  e.  A )
Assertion
Ref Expression
bnj1468  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Distinct variable groups:    y, A    y, V    ph, y    ps, y    x, y
Allowed substitution hints:    ph( x)    ps( x)    A( x)    V( x)

Proof of Theorem bnj1468
StepHypRef Expression
1 sbcco 3119 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph )
2 ax-17 1623 . . 3  |-  ( ps 
->  A. y ps )
3 bnj1468.3 . . . . . . . 8  |-  ( y  e.  A  ->  A. x  y  e.  A )
43nfcii 2507 . . . . . . 7  |-  F/_ x A
54nfeq2 2527 . . . . . 6  |-  F/ x  y  =  A
6 nfsbc1v 3116 . . . . . . 7  |-  F/ x [. y  /  x ]. ph
7 bnj1468.1 . . . . . . . 8  |-  ( ps 
->  A. x ps )
87nfi 1557 . . . . . . 7  |-  F/ x ps
96, 8nfbi 1846 . . . . . 6  |-  F/ x
( [. y  /  x ]. ph  <->  ps )
105, 9nfim 1822 . . . . 5  |-  F/ x
( y  =  A  ->  ( [. y  /  x ]. ph  <->  ps )
)
1110nfri 1770 . . . 4  |-  ( ( y  =  A  -> 
( [. y  /  x ]. ph  <->  ps ) )  ->  A. x ( y  =  A  ->  ( [. y  /  x ]. ph  <->  ps )
) )
12 a9ev 1663 . . . . 5  |-  E. x  x  =  y
13 eqeq1 2386 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
14 bnj1468.2 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
1513, 14syl6bir 221 . . . . . 6  |-  ( x  =  y  ->  (
y  =  A  -> 
( ph  <->  ps ) ) )
16 sbceq1a 3107 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<-> 
[. y  /  x ]. ph ) )
1716bibi1d 311 . . . . . 6  |-  ( x  =  y  ->  (
( ph  <->  ps )  <->  ( [. y  /  x ]. ph  <->  ps )
) )
1815, 17sylibd 206 . . . . 5  |-  ( x  =  y  ->  (
y  =  A  -> 
( [. y  /  x ]. ph  <->  ps ) ) )
1912, 18bnj101 28419 . . . 4  |-  E. x
( y  =  A  ->  ( [. y  /  x ]. ph  <->  ps )
)
2011, 19bnj1131 28489 . . 3  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  ps ) )
212, 20bnj1464 28546 . 2  |-  ( A  e.  V  ->  ( [. A  /  y ]. [. y  /  x ]. ph  <->  ps ) )
221, 21syl5bbr 251 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1546    = wceq 1649    e. wcel 1717   [.wsbc 3097
This theorem is referenced by:  bnj1463  28755
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
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-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-sbc 3098
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