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Theorem bj-sbftv 31344
Description: Version of sbft 2177 with a dv condition, which does not require ax-13 2057. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-sbftv  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem bj-sbftv
StepHypRef Expression
1 spsbe 1794 . . 3  |-  ( [ y  /  x ] ph  ->  E. x ph )
2 19.9t 1946 . . 3  |-  ( F/ x ph  ->  ( E. x ph  <->  ph ) )
31, 2syl5ib 222 . 2  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  ->  ph ) )
4 nfr 1928 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
5 bj-stdpc4v 31333 . . 3  |-  ( A. x ph  ->  [ y  /  x ] ph )
64, 5syl6 34 . 2  |-  ( F/ x ph  ->  ( ph  ->  [ y  /  x ] ph ) )
73, 6impbid 193 1  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435   E.wex 1657   F/wnf 1661   [wsb 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791
This theorem is referenced by:  bj-sbfv  31345
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