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Theorem sbciegft 3034
Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3035.) (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Assertion
Ref Expression
sbciegft  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  <->  ps ) )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)    V( x)

Proof of Theorem sbciegft
StepHypRef Expression
1 sbc5 3028 . . 3  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
2 bi1 178 . . . . . . . 8  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
32imim2i 13 . . . . . . 7  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ph  ->  ps ) ) )
43imp3a 420 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ( x  =  A  /\  ph )  ->  ps ) )
54alimi 1549 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( ( x  =  A  /\  ph )  ->  ps ) )
6 19.23t 1808 . . . . . 6  |-  ( F/ x ps  ->  ( A. x ( ( x  =  A  /\  ph )  ->  ps )  <->  ( E. x ( x  =  A  /\  ph )  ->  ps ) ) )
76biimpa 470 . . . . 5  |-  ( ( F/ x ps  /\  A. x ( ( x  =  A  /\  ph )  ->  ps ) )  ->  ( E. x
( x  =  A  /\  ph )  ->  ps ) )
85, 7sylan2 460 . . . 4  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( E. x ( x  =  A  /\  ph )  ->  ps )
)
983adant1 973 . . 3  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( E. x ( x  =  A  /\  ph )  ->  ps )
)
101, 9syl5bi 208 . 2  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  ->  ps )
)
11 bi2 189 . . . . . . . 8  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
1211imim2i 13 . . . . . . 7  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ps  ->  ph ) ) )
1312com23 72 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ps  ->  (
x  =  A  ->  ph ) ) )
1413alimi 1549 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( ps  ->  (
x  =  A  ->  ph ) ) )
15 19.21t 1802 . . . . . 6  |-  ( F/ x ps  ->  ( A. x ( ps  ->  ( x  =  A  ->  ph ) )  <->  ( ps  ->  A. x ( x  =  A  ->  ph )
) ) )
1615biimpa 470 . . . . 5  |-  ( ( F/ x ps  /\  A. x ( ps  ->  ( x  =  A  ->  ph ) ) )  -> 
( ps  ->  A. x
( x  =  A  ->  ph ) ) )
1714, 16sylan2 460 . . . 4  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( ps  ->  A. x
( x  =  A  ->  ph ) ) )
18173adant1 973 . . 3  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( ps  ->  A. x
( x  =  A  ->  ph ) ) )
19 sbc6g 3029 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
20193ad2ant1 976 . . 3  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
2118, 20sylibrd 225 . 2  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( ps  ->  [. A  /  x ]. ph )
)
2210, 21impbid 183 1  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530   E.wex 1531   F/wnf 1534    = wceq 1632    e. wcel 1696   [.wsbc 3004
This theorem is referenced by:  sbciegf  3035  sbciedf  3039
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
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-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sbc 3005
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