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Theorem sblim 1931
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 1927 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 sblim.1 . . . 4  |-  F/ x ps
32sbf 1751 . . 3  |-  ( [ y  /  x ] ps 
<->  ps )
43imbi2i 225 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  <->  ( [ y  /  x ] ph  ->  ps )
)
51, 4bitri 183 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   F/wnf 1437   [wsb 1736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737
This theorem is referenced by:  sbnf2  1957  sbmo  2059
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