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Theorem sbft 2081
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, 22-Apr-2018.)
Assertion
Ref Expression
sbft  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )

Proof of Theorem sbft
StepHypRef Expression
1 sb1 1659 . . . 4  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
2 simpr 448 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  ph )
32eximi 1582 . . . 4  |-  ( E. x ( x  =  y  /\  ph )  ->  E. x ph )
41, 3syl 16 . . 3  |-  ( [ y  /  x ] ph  ->  E. x ph )
5 19.9t 1789 . . 3  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
64, 5syl5ib 211 . 2  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  ->  ph ) )
7 nfr 1773 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
8 stdpc4 2080 . . 3  |-  ( A. x ph  ->  [ y  /  x ] ph )
97, 8syl6 31 . 2  |-  ( F/ x ph  ->  ( ph  ->  [ y  /  x ] ph ) )
106, 9impbid 184 1  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547   F/wnf 1550   [wsb 1655
This theorem is referenced by:  sbf  2083  sbctt  3191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-nf 1551  df-sb 1656
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