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Theorem sbiedv 1977
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 1978). (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 1605 . 2  |-  F/ x ph
2 nfvd 1606 . 2  |-  ( ph  ->  F/ x ch )
3 sbiedv.1 . . 3  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
43ex 423 . 2  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
51, 2, 4sbied 1976 1  |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   [wsb 1629
This theorem is referenced by:  iscatd2  13583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-sb 1630
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