ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbciegft Unicode version

Theorem sbciegft 2993
Description: Conversion of implicit substitution to explicit class substitution, using a bound-variable hypothesis instead of distinct variables. (Closed theorem version of sbciegf 2994.) (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 2986 . . 3  |-  ( [. A  /  x ]. ph  <->  E. x
( x  =  A  /\  ph ) )
2 biimp 118 . . . . . . . 8  |-  ( (
ph 
<->  ps )  ->  ( ph  ->  ps ) )
32imim2i 12 . . . . . . 7  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( x  =  A  ->  ( ph  ->  ps ) ) )
43impd 254 . . . . . 6  |-  ( ( x  =  A  -> 
( ph  <->  ps ) )  -> 
( ( x  =  A  /\  ph )  ->  ps ) )
54alimi 1455 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( ( x  =  A  /\  ph )  ->  ps ) )
6 19.23t 1677 . . . . . 6  |-  ( F/ x ps  ->  ( A. x ( ( x  =  A  /\  ph )  ->  ps )  <->  ( E. x ( x  =  A  /\  ph )  ->  ps ) ) )
76biimpa 296 . . . . 5  |-  ( ( F/ x ps  /\  A. x ( ( x  =  A  /\  ph )  ->  ps ) )  ->  ( E. x
( x  =  A  /\  ph )  ->  ps ) )
85, 7sylan2 286 . . . 4  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( E. x ( x  =  A  /\  ph )  ->  ps )
)
983adant1 1015 . . 3  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( E. x ( x  =  A  /\  ph )  ->  ps )
)
101, 9biimtrid 152 . 2  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  ->  ps )
)
11 biimpr 130 . . . . . . . 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 1455 . . . . 5  |-  ( A. x ( x  =  A  ->  ( ph  <->  ps ) )  ->  A. x
( ps  ->  (
x  =  A  ->  ph ) ) )
15 19.21t 1582 . . . . . 6  |-  ( F/ x ps  ->  ( A. x ( ps  ->  ( x  =  A  ->  ph ) )  <->  ( ps  ->  A. x ( x  =  A  ->  ph )
) ) )
1615biimpa 296 . . . . 5  |-  ( ( F/ x ps  /\  A. x ( ps  ->  ( x  =  A  ->  ph ) ) )  -> 
( ps  ->  A. x
( x  =  A  ->  ph ) ) )
1714, 16sylan2 286 . . . 4  |-  ( ( F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( ps  ->  A. x
( x  =  A  ->  ph ) ) )
18173adant1 1015 . . 3  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( ps  ->  A. x
( x  =  A  ->  ph ) ) )
19 sbc6g 2987 . . . 4  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
20193ad2ant1 1018 . . 3  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
2118, 20sylibrd 169 . 2  |-  ( ( A  e.  V  /\  F/ x ps  /\  A. x ( x  =  A  ->  ( ph  <->  ps ) ) )  -> 
( ps  ->  [. A  /  x ]. ph )
)
2210, 21impbid 129 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 104    <-> wb 105    /\ w3a 978   A.wal 1351    = wceq 1353   F/wnf 1460   E.wex 1492    e. wcel 2148   [.wsbc 2962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-sbc 2963
This theorem is referenced by:  sbciegf  2994  sbciedf  2998
  Copyright terms: Public domain W3C validator