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Theorem sbrim 2101
Description: Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sbrim.1  |-  F/ x ph
Assertion
Ref Expression
sbrim  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )

Proof of Theorem sbrim
StepHypRef Expression
1 sbim 2099 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 sbrim.1 . . . 4  |-  F/ x ph
32sbf 2060 . . 3  |-  ( [ y  /  x ] ph 
<-> 
ph )
43imbi1i 316 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
51, 4bitri 241 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   F/wnf 1550   [wsb 1655
This theorem is referenced by:  sbco2d  2121
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
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