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Theorem sbciegft 3355
Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 3356.) (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 3349 . . 3  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
2 bi1 186 . . . . . . . 8  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
32imim2i 14 . . . . . . 7  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ph  ->  ps ) ) )
43impd 429 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ( x  =  A  /\  ph )  ->  ps ) )
54alimi 1638 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( ( x  =  A  /\  ph )  ->  ps ) )
6 19.23t 1914 . . . . . 6  |-  ( F/ x ps  ->  ( A. x ( ( x  =  A  /\  ph )  ->  ps )  <->  ( E. x ( x  =  A  /\  ph )  ->  ps ) ) )
76biimpa 482 . . . . 5  |-  ( ( F/ x ps  /\  A. x ( ( x  =  A  /\  ph )  ->  ps ) )  ->  ( E. x
( x  =  A  /\  ph )  ->  ps ) )
85, 7sylan2 472 . . . 4  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( E. x ( x  =  A  /\  ph )  ->  ps )
)
983adant1 1012 . . 3  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( E. x ( x  =  A  /\  ph )  ->  ps )
)
101, 9syl5bi 217 . 2  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  ->  ps )
)
11 bi2 198 . . . . . . . 8  |-  ( (
ph 
<->  ps )  ->  ( ps  ->  ph ) )
1211imim2i 14 . . . . . . 7  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ps  ->  ph ) ) )
1312com23 78 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ps  ->  (
x  =  A  ->  ph ) ) )
1413alimi 1638 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( ps  ->  (
x  =  A  ->  ph ) ) )
15 19.21t 1909 . . . . . 6  |-  ( F/ x ps  ->  ( A. x ( ps  ->  ( x  =  A  ->  ph ) )  <->  ( ps  ->  A. x ( x  =  A  ->  ph )
) ) )
1615biimpa 482 . . . . 5  |-  ( ( F/ x ps  /\  A. x ( ps  ->  ( x  =  A  ->  ph ) ) )  -> 
( ps  ->  A. x
( x  =  A  ->  ph ) ) )
1714, 16sylan2 472 . . . 4  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( ps  ->  A. x
( x  =  A  ->  ph ) ) )
18173adant1 1012 . . 3  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( ps  ->  A. x
( x  =  A  ->  ph ) ) )
19 sbc6g 3350 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
20193ad2ant1 1015 . . 3  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
2118, 20sylibrd 234 . 2  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( ps  ->  [. A  /  x ]. ph )
)
2210, 21impbid 191 1  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971   A.wal 1396    = wceq 1398   E.wex 1617   F/wnf 1621    e. wcel 1823   [.wsbc 3324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3108  df-sbc 3325
This theorem is referenced by:  sbciegf  3356  sbciedf  3360
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