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Theorem sbied 1761
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1764). (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 1499 . 2  |-  ( ph  ->  A. x ph )
3 sbied.2 . . 3  |-  ( ph  ->  F/ x ch )
43nfrd 1500 . 2  |-  ( ph  ->  ( ch  ->  A. x ch ) )
5 sbied.3 . 2  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
62, 4, 5sbiedh 1760 1  |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   F/wnf 1436   [wsb 1735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  sbiedv  1762  dvelimdf  1989  cbvrald  12984
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