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Theorem hbsb3 1983
Description: If  y is not free in  ph,  x is not free in  [ y  /  x ] ph. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
hbsb3.1  |-  ( ph  ->  A. y ph )
Assertion
Ref Expression
hbsb3  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )

Proof of Theorem hbsb3
StepHypRef Expression
1 hbsb3.1 . . 3  |-  ( ph  ->  A. y ph )
21sbimi 1633 . 2  |-  ( [ y  /  x ] ph  ->  [ y  /  x ] A. y ph )
3 hbsb2a 1981 . 2  |-  ( [ y  /  x ] A. y ph  ->  A. x [ y  /  x ] ph )
42, 3syl 15 1  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527   [wsb 1629
This theorem is referenced by:  nfs1  1984  ax16ALT  1987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-sb 1630
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