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Theorem sbim 1958
Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbim  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)

Proof of Theorem sbim
StepHypRef Expression
1 sbi1 1956 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
2 sbi2 1957 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
31, 2impbii 182 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178   [wsb 1883
This theorem is referenced by:  sbor  1959  sbrim  1960  sblim  1961  sban  1962  sbbi  1964  sbequ8  1972  sbcimg  2993
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884
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