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Theorem sbied 2152
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 2151). (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Jun-2018.)
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 sbied.1 . . . 4  |-  F/ x ph
21sbrim 2138 . . 3  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
3 sbied.2 . . . . 5  |-  ( ph  ->  F/ x ch )
41, 3nfim1 1831 . . . 4  |-  F/ x
( ph  ->  ch )
5 sbied.3 . . . . . 6  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
65com12 30 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  ( ps  <->  ch )
) )
76pm5.74d 240 . . . 4  |-  ( x  =  y  ->  (
( ph  ->  ps )  <->  (
ph  ->  ch ) ) )
84, 7sbie 2151 . . 3  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  ch ) )
92, 8bitr3i 244 . 2  |-  ( (
ph  ->  [ y  /  x ] ps )  <->  ( ph  ->  ch ) )
109pm5.74ri 239 1  |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   F/wnf 1554   [wsb 1659
This theorem is referenced by:  sbieOLD  2154  sbiedv  2155  dvelimdfOLD  2159  sbco2  2163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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