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Theorem sbft 2079
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 1706 . . 3  |-  ( [ y  /  x ] ph  ->  E. x ph )
2 19.9t 1828 . . 3  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
31, 2syl5ib 219 . 2  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  ->  ph ) )
4 nfr 1811 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
5 stdpc4 2053 . . 3  |-  ( A. x ph  ->  [ y  /  x ] ph )
64, 5syl6 33 . 2  |-  ( F/ x ph  ->  ( ph  ->  [ y  /  x ] ph ) )
73, 6impbid 191 1  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1368   E.wex 1587   F/wnf 1590   [wsb 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-13 1954
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-sb 1703
This theorem is referenced by:  sbf  2080  sbctt  3359  wl-sbrimt  28517  wl-sblimt  28518  wl-equsb4  28524
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