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Theorem sbciegft 2909
Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2910.) (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 2903 . . 3  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
2 bi1 117 . . . . . . . 8  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
32imim2i 12 . . . . . . 7  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ph  ->  ps ) ) )
43impd 252 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ( x  =  A  /\  ph )  ->  ps ) )
54alimi 1414 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( ( x  =  A  /\  ph )  ->  ps ) )
6 19.23t 1638 . . . . . 6  |-  ( F/ x ps  ->  ( A. x ( ( x  =  A  /\  ph )  ->  ps )  <->  ( E. x ( x  =  A  /\  ph )  ->  ps ) ) )
76biimpa 292 . . . . 5  |-  ( ( F/ x ps  /\  A. x ( ( x  =  A  /\  ph )  ->  ps ) )  ->  ( E. x
( x  =  A  /\  ph )  ->  ps ) )
85, 7sylan2 282 . . . 4  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( E. x ( x  =  A  /\  ph )  ->  ps )
)
983adant1 982 . . 3  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( E. x ( x  =  A  /\  ph )  ->  ps )
)
101, 9syl5bi 151 . 2  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  ->  ps )
)
11 bi2 129 . . . . . . . 8  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
1211imim2i 12 . . . . . . 7  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ps  ->  ph ) ) )
1312com23 78 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ps  ->  (
x  =  A  ->  ph ) ) )
1413alimi 1414 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( ps  ->  (
x  =  A  ->  ph ) ) )
15 19.21t 1544 . . . . . 6  |-  ( F/ x ps  ->  ( A. x ( ps  ->  ( x  =  A  ->  ph ) )  <->  ( ps  ->  A. x ( x  =  A  ->  ph )
) ) )
1615biimpa 292 . . . . 5  |-  ( ( F/ x ps  /\  A. x ( ps  ->  ( x  =  A  ->  ph ) ) )  -> 
( ps  ->  A. x
( x  =  A  ->  ph ) ) )
1714, 16sylan2 282 . . . 4  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( ps  ->  A. x
( x  =  A  ->  ph ) ) )
18173adant1 982 . . 3  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( ps  ->  A. x
( x  =  A  ->  ph ) ) )
19 sbc6g 2904 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
20193ad2ant1 985 . . 3  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
2118, 20sylibrd 168 . 2  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( ps  ->  [. A  /  x ]. ph )
)
2210, 21impbid 128 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    /\ wa 103    <-> wb 104    /\ w3a 945   A.wal 1312    = wceq 1314   F/wnf 1419   E.wex 1451    e. wcel 1463   [.wsbc 2880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-sbc 2881
This theorem is referenced by:  sbciegf  2910  sbciedf  2914
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