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Theorem wl-sbrimt 31922
Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2235. (Contributed by Wolf Lammen, 26-Jul-2019.)
Assertion
Ref Expression
wl-sbrimt  |-  ( F/ x ph  ->  ( [ y  /  x ] ( ph  ->  ps )  <->  ( ph  ->  [ y  /  x ] ps ) ) )

Proof of Theorem wl-sbrimt
StepHypRef Expression
1 sbim 2234 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 sbft 2218 . . 3  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
32imbi1d 323 . 2  |-  ( F/ x ph  ->  (
( [ y  /  x ] ph  ->  [ y  /  x ] ps ) 
<->  ( ph  ->  [ y  /  x ] ps ) ) )
41, 3syl5bb 265 1  |-  ( F/ x ph  ->  ( [ 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: (None)
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