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Theorem sbhb 2234
Description: Two ways of expressing " x is (effectively) not free in  ph." (Contributed by NM, 29-May-2009.)
Assertion
Ref Expression
sbhb  |-  ( (
ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) )
Distinct variable group:    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem sbhb
StepHypRef Expression
1 nfv 1754 . . . 4  |-  F/ y
ph
21sb8 2219 . . 3  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
32imbi2i 313 . 2  |-  ( (
ph  ->  A. x ph )  <->  (
ph  ->  A. y [ y  /  x ] ph ) )
4 19.21v 1778 . 2  |-  ( A. y ( ph  ->  [ y  /  x ] ph )  <->  ( ph  ->  A. y [ y  /  x ] ph ) )
53, 4bitr4i 255 1  |-  ( (
ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187   A.wal 1435   [wsb 1789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1660  df-nf 1664  df-sb 1790
This theorem is referenced by: (None)
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