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Theorem sbim 2193
Description: Implication inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
sbim  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)

Proof of Theorem sbim
StepHypRef Expression
1 sbi1 2190 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
2 sbi2 2191 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
31, 2impbii 190 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   [wsb 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791
This theorem is referenced by:  sbrim  2194  sblim  2195  sbor  2196  sban  2197  sbbi  2199  sbequ8ALT  2205  sbcimg  3341  mo5f  28118  iuninc  28178  suppss2fOLD  28239  suppss2f  28240  esumpfinvalf  28905  bj-sbnf  31411  wl-sbrimt  31842  wl-sblimt  31843  frege58bcor  36469  frege60b  36471  frege65b  36476  ellimcabssub0  37637
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