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Theorem sbrim 2138
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 2137 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 sbrim.1 . . . 4  |-  F/ x ph
32sbf 2119 . . 3  |-  ( [ y  /  x ] ph 
<-> 
ph )
43imbi1i 317 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
51, 4bitri 242 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   F/wnf 1554   [wsb 1659
This theorem is referenced by:  sbiedALT  2154  sbco2d  2164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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