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Theorem sbiedv 2153
Description: Conversion of implicit substitution to explicit substitution (deduction version of sbie 2149). (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 1629 . 2  |-  F/ x ph
2 nfvd 1630 . 2  |-  ( ph  ->  F/ x ch )
3 sbiedv.1 . . 3  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
43ex 424 . 2  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
51, 2, 4sbied 2150 1  |-  ( ph  ->  ( [ y  /  x ] ps  <->  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   [wsb 1658
This theorem is referenced by:  2mos  2360  iscatd2  13906  prtlem5  26705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
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