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Theorem hbsb 2186
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 1560 . . 3  |-  F/ z
ph
32nfsb 2185 . 2  |-  F/ z [ y  /  x ] ph
43nfri 1778 1  |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549   [wsb 1658
This theorem is referenced by:  hbab  2427  hblem  2540
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-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
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