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Theorem sbhb 1859
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 ax-17 1460 . . . 4  |-  ( ph  ->  A. y ph )
21sb8h 1777 . . 3  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
32imbi2i 224 . 2  |-  ( (
ph  ->  A. x ph )  <->  (
ph  ->  A. y [ y  /  x ] ph ) )
4 19.21v 1796 . 2  |-  ( A. y ( ph  ->  [ y  /  x ] ph )  <->  ( ph  ->  A. y [ y  /  x ] ph ) )
53, 4bitr4i 185 1  |-  ( (
ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   [wsb 1687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by:  sbnf2  1900
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