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Theorem sbft 1773
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 1767 . . 3  |-  ( [ y  /  x ] ph  ->  E. x ph )
2 19.9t 1576 . . 3  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
31, 2syl5ib 152 . 2  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  ->  ph ) )
4 nfr 1454 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
5 stdpc4 1702 . . 3  |-  ( A. x ph  ->  [ y  /  x ] ph )
64, 5syl6 33 . 2  |-  ( F/ x ph  ->  ( ph  ->  [ y  /  x ] ph ) )
73, 6impbid 127 1  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1285   F/wnf 1392   E.wex 1424   [wsb 1689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-i9 1466  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690
This theorem is referenced by:  sbctt  2894
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