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Theorem wl-sbhbt 31882
Description: Closed form of sbhb 2267. Characterizing the expression 
ph  ->  A. x ph using a substitution expression. (Contributed by Wolf Lammen, 28-Jul-2019.)
Assertion
Ref Expression
wl-sbhbt  |-  ( A. x F/ y ph  ->  ( ( ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) ) )

Proof of Theorem wl-sbhbt
StepHypRef Expression
1 wl-sb8t 31880 . . 3  |-  ( A. x F/ y ph  ->  ( A. x ph  <->  A. y [ y  /  x ] ph ) )
21imbi2d 318 . 2  |-  ( A. x F/ y ph  ->  ( ( ph  ->  A. x ph )  <->  ( ph  ->  A. y [ y  /  x ] ph ) ) )
3 19.21t 1986 . . 3  |-  ( F/ y ph  ->  ( A. y ( ph  ->  [ y  /  x ] ph )  <->  ( ph  ->  A. y [ y  /  x ] ph ) ) )
43sps 1943 . 2  |-  ( A. x F/ y ph  ->  ( A. y ( ph  ->  [ y  /  x ] ph )  <->  ( ph  ->  A. y [ y  /  x ] ph ) ) )
52, 4bitr4d 260 1  |-  ( A. x F/ y ph  ->  ( ( ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1442   F/wnf 1667   [wsb 1797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668  df-sb 1798
This theorem is referenced by:  wl-sbnf1  31883
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