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Theorem sbhb 2183
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 1629 . . . 4  |-  F/ y
ph
21sb8 2168 . . 3  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
32imbi2i 304 . 2  |-  ( (
ph  ->  A. x ph )  <->  (
ph  ->  A. y [ y  /  x ] ph ) )
4 19.21v 1913 . 2  |-  ( A. y ( ph  ->  [ y  /  x ] ph )  <->  ( ph  ->  A. y [ y  /  x ] ph ) )
53, 4bitr4i 244 1  |-  ( (
ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   [wsb 1658
This theorem is referenced by:  sbnf2  2184
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659
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