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Theorem sbiedv 1762
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1764). (Contributed by NM, 7-Jan-2017.)
Hypothesis
Ref Expression
sbiedv.1  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
sbiedv  |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch )
)
Distinct variable groups:    ph, x    ch, x
Allowed substitution hints:    ph( y)    ps( x, y)    ch( y)

Proof of Theorem sbiedv
StepHypRef Expression
1 nfv 1508 . 2  |-  F/ x ph
2 nfvd 1509 . 2  |-  ( ph  ->  F/ x ch )
3 sbiedv.1 . . 3  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
43ex 114 . 2  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
51, 2, 4sbied 1761 1  |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   [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-17 1506  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:  acexmid  5766
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