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Theorem sbim 2235
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 2232 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
2 sbi2 2233 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
31, 2impbii 192 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   [wsb 1808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-12 1944  ax-13 2102
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1675  df-nf 1679  df-sb 1809
This theorem is referenced by:  sbrim  2236  sblim  2237  sbor  2238  sban  2239  sbbi  2241  sbequ8ALT  2247  sbcimg  3321  mo5f  28176  iuninc  28230  suppss2fOLD  28290  suppss2f  28291  esumpfinvalf  28948  bj-sbnf  31487  wl-sbrimt  31924  wl-sblimt  31925  frege58bcor  36545  frege60b  36547  frege65b  36552  ellimcabssub0  37783
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