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Theorem sbft 2117
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 1664 . . 3  |-  ( [ y  /  x ] ph  ->  E. x ph )
2 19.9t 1794 . . 3  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
31, 2syl5ib 212 . 2  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  ->  ph ) )
4 nfr 1778 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
5 stdpc4 2092 . . 3  |-  ( A. x ph  ->  [ y  /  x ] ph )
64, 5syl6 32 . 2  |-  ( F/ x ph  ->  ( ph  ->  [ y  /  x ] ph ) )
73, 6impbid 185 1  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   A.wal 1550   E.wex 1551   F/wnf 1554   [wsb 1659
This theorem is referenced by:  sbf  2119  sbctt  3225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-nf 1555  df-sb 1660
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