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Theorem hbsb 2270
Description: If  z is not free in  ph, it is not free in  [ y  /  x ] ph when  y and  z are distinct. (Contributed by NM, 12-Aug-1993.)
Hypothesis
Ref Expression
hbsb.1  |-  ( ph  ->  A. z ph )
Assertion
Ref Expression
hbsb  |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
Distinct variable group:    y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem hbsb
StepHypRef Expression
1 hbsb.1 . . . 4  |-  ( ph  ->  A. z ph )
21nfi 1674 . . 3  |-  F/ z
ph
32nfsb 2269 . 2  |-  F/ z [ y  /  x ] ph
43nfri 1952 1  |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1442   [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:  hbab  2442  hblem  2559  bj-hblem  31458
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