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Theorem sbied 1718
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1721). (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 sbied.1 . . 3  |-  F/ x ph
21nfri 1457 . 2  |-  ( ph  ->  A. x ph )
3 sbied.2 . . 3  |-  ( ph  ->  F/ x ch )
43nfrd 1458 . 2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
5 sbied.3 . 2  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
62, 4, 5sbiedh 1717 1  |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   F/wnf 1394   [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  sbiedv  1719  dvelimdf  1940  cbvrald  11334
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