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Theorem sbim 2114
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 2112 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
2 sbi2 2113 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
31, 2impbii 181 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   [wsb 1655
This theorem is referenced by:  sbor  2115  sbrim  2116  sblim  2117  sban  2118  sbbi  2120  sbequ8  2128  sbcimg  3162  mo5f  23925  iuninc  23964  suppss2f  24002  esumpfinvalf  24419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
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