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Theorem sbied 1989
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1991). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypotheses
Ref Expression
sbied.1  |-  F/ x ph
sbied.2  |-  ( ph  ->  F/ x ch )
sbied.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
sbied  |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch )
)

Proof of Theorem sbied
StepHypRef Expression
1 sb1 1641 . . . 4  |-  ( [ y  /  x ] ps  ->  E. x ( x  =  y  /\  ps ) )
2 sbied.1 . . . . 5  |-  F/ x ph
3 sbied.3 . . . . . . 7  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
4 bi1 178 . . . . . . 7  |-  ( ( ps  <->  ch )  ->  ( ps  ->  ch ) )
53, 4syl6 29 . . . . . 6  |-  ( ph  ->  ( x  =  y  ->  ( ps  ->  ch ) ) )
65imp3a 420 . . . . 5  |-  ( ph  ->  ( ( x  =  y  /\  ps )  ->  ch ) )
72, 6eximd 1762 . . . 4  |-  ( ph  ->  ( E. x ( x  =  y  /\  ps )  ->  E. x ch ) )
81, 7syl5 28 . . 3  |-  ( ph  ->  ( [ y  /  x ] ps  ->  E. x ch ) )
9 sbied.2 . . . 4  |-  ( ph  ->  F/ x ch )
10919.9d 1796 . . 3  |-  ( ph  ->  ( E. x ch 
->  ch ) )
118, 10syld 40 . 2  |-  ( ph  ->  ( [ y  /  x ] ps  ->  ch ) )
129nfrd 1755 . . 3  |-  ( ph  ->  ( ch  ->  A. x ch ) )
13 bi2 189 . . . . . . 7  |-  ( ( ps  <->  ch )  ->  ( ch  ->  ps ) )
143, 13syl6 29 . . . . . 6  |-  ( ph  ->  ( x  =  y  ->  ( ch  ->  ps ) ) )
1514com23 72 . . . . 5  |-  ( ph  ->  ( ch  ->  (
x  =  y  ->  ps ) ) )
162, 15alimd 1756 . . . 4  |-  ( ph  ->  ( A. x ch 
->  A. x ( x  =  y  ->  ps ) ) )
17 sb2 1976 . . . 4  |-  ( A. x ( x  =  y  ->  ps )  ->  [ y  /  x ] ps )
1816, 17syl6 29 . . 3  |-  ( ph  ->  ( A. x ch 
->  [ y  /  x ] ps ) )
1912, 18syld 40 . 2  |-  ( ph  ->  ( ch  ->  [ y  /  x ] ps ) )
2011, 19impbid 183 1  |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531   F/wnf 1534   [wsb 1638
This theorem is referenced by:  sbiedv  1990  sbie  1991  dvelimdf  2035  sbco2  2039  prtlem5  26825
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
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-sb 1639
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