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Theorem sbhb 2046
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 1605 . . . 4  |-  F/ y
ph
21sb8 2032 . . 3  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
32imbi2i 303 . 2  |-  ( (
ph  ->  A. x ph )  <->  (
ph  ->  A. y [ y  /  x ] ph ) )
4 19.21v 1831 . 2  |-  ( A. y ( ph  ->  [ y  /  x ] ph )  <->  ( ph  ->  A. y [ y  /  x ] ph ) )
53, 4bitr4i 243 1  |-  ( (
ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   [wsb 1629
This theorem is referenced by:  sbnf2  2047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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