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Theorem sbbid 2166
Description: Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.)
Hypotheses
Ref Expression
sbbid.1  |-  F/ x ph
sbbid.2  |-  ( ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
sbbid  |-  ( ph  ->  ( [ y  /  x ] ps  <->  [ y  /  x ] ch )
)

Proof of Theorem sbbid
StepHypRef Expression
1 sbbid.1 . . 3  |-  F/ x ph
2 sbbid.2 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
31, 2alrimi 1899 . 2  |-  ( ph  ->  A. x ( ps  <->  ch ) )
4 spsbbi 2165 . 2  |-  ( A. x ( ps  <->  ch )  ->  ( [ y  /  x ] ps  <->  [ y  /  x ] ch )
)
53, 4syl 17 1  |-  ( ph  ->  ( [ y  /  x ] ps  <->  [ y  /  x ] ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1401   F/wnf 1635   [wsb 1761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-12 1876  ax-13 2024
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-ex 1632  df-nf 1636  df-sb 1762
This theorem is referenced by:  sbcom3  2175  sbco3  2182  sbcom2  2211  sbal  2228  wl-equsb3  31340  wl-sbcom2d-lem1  31345  wl-sbcom3  31371
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