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Theorem hbsb 2062
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 1541 . . 3  |-  F/ z
ph
32nfsb 2061 . 2  |-  F/ z [ y  /  x ] ph
43nfri 1754 1  |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530   [wsb 1638
This theorem is referenced by:  hbab  2287  hblem  2400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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