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Theorem hbsb 1936
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.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.)
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 1449 . . 3  |-  F/ z
ph
32nfsb 1933 . 2  |-  F/ z [ y  /  x ] ph
43nfri 1506 1  |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1340   [wsb 1749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522
This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750
This theorem is referenced by:  sb10f  1982  hbsb4  1999  sb8euh  2036  hbab  2155  hblem  2272
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