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Theorem sblim 2021
Description: Substitution with a variable not free in consequent affects only the antecedent. (Contributed by NM, 14-Nov-2013.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sblim.1  |-  F/ x ps
Assertion
Ref Expression
sblim  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps )
)

Proof of Theorem sblim
StepHypRef Expression
1 sbim 2018 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 sblim.1 . . . 4  |-  F/ x ps
32sbf 1979 . . 3  |-  ( [ y  /  x ] ps 
<->  ps )
43imbi2i 303 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  <->  ( [ y  /  x ] ph  ->  ps )
)
51, 4bitri 240 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1534   [wsb 1638
This theorem is referenced by:  sbnf2  2060  sbmo  2186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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