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Theorem sbft 2218
Description: Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)
Assertion
Ref Expression
sbft  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )

Proof of Theorem sbft
StepHypRef Expression
1 spsbe 1811 . . 3  |-  ( [ y  /  x ] ph  ->  E. x ph )
2 19.9t 1979 . . 3  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
31, 2syl5ib 227 . 2  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  ->  ph ) )
4 nfr 1961 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
5 stdpc4 2194 . . 3  |-  ( A. x ph  ->  [ y  /  x ] ph )
64, 5syl6 34 . 2  |-  ( F/ x ph  ->  ( ph  ->  [ y  /  x ] ph ) )
73, 6impbid 195 1  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1452   E.wex 1673   F/wnf 1677   [wsb 1807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808
This theorem is referenced by:  sbf  2219  sbctt  3341  wl-sbrimt  31922  wl-sblimt  31923  wl-equsb4  31929
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