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Theorem sbrim 2235
Description: Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 2-Jun-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 2234 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 sbrim.1 . . . 4  |-  F/ x ph
32sbf 2219 . . 3  |-  ( [ y  /  x ] ph 
<-> 
ph )
43imbi1i 331 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
51, 4bitri 257 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   F/wnf 1677   [wsb 1807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808
This theorem is referenced by:  sbied  2248  sbco2d  2255
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