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Theorem wl-sblimt 31879
Description: Substitution with a variable not free in antecedent affects only the consequent. Closed form of sbrim 2225. (Contributed by Wolf Lammen, 26-Jul-2019.)
Assertion
Ref Expression
wl-sblimt  |-  ( F/ x ps  ->  ( [ y  /  x ] ( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps ) ) )

Proof of Theorem wl-sblimt
StepHypRef Expression
1 sbim 2224 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 sbft 2208 . . 3  |-  ( F/ x ps  ->  ( [ y  /  x ] ps  <->  ps ) )
32imbi2d 318 . 2  |-  ( F/ x ps  ->  (
( [ y  /  x ] ph  ->  [ y  /  x ] ps ) 
<->  ( [ y  /  x ] ph  ->  ps ) ) )
41, 3syl5bb 261 1  |-  ( F/ x ps  ->  ( [ y  /  x ] ( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   F/wnf 1667   [wsb 1797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-ex 1664  df-nf 1668  df-sb 1798
This theorem is referenced by: (None)
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