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Theorem sbrim 2141
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 2121 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 sbrim.1 . . . 4  |-  F/ x ph
32sbf 2105 . . 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 1553   [wsb 1658
This theorem is referenced by:  sbco2d  2162
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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