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Theorem hbim 1725
Description: If  x is not free in  ph and  ps, it is not free in  ( ph  ->  ps ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.)
Hypotheses
Ref Expression
hbim.1  |-  ( ph  ->  A. x ph )
hbim.2  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
hbim  |-  ( (
ph  ->  ps )  ->  A. x ( ph  ->  ps ) )

Proof of Theorem hbim
StepHypRef Expression
1 hbim.1 . . . 4  |-  ( ph  ->  A. x ph )
21hbn 1720 . . 3  |-  ( -. 
ph  ->  A. x  -.  ph )
3 pm2.21 100 . . 3  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
42, 3alrimih 1552 . 2  |-  ( -. 
ph  ->  A. x ( ph  ->  ps ) )
5 hbim.2 . . 3  |-  ( ps 
->  A. x ps )
6 ax-1 5 . . 3  |-  ( ps 
->  ( ph  ->  ps ) )
75, 6alrimih 1552 . 2  |-  ( ps 
->  A. x ( ph  ->  ps ) )
84, 7ja 153 1  |-  ( (
ph  ->  ps )  ->  A. x ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1527
This theorem is referenced by:  19.23h  1728  19.21h  1731  hban  1736  cbv3hv  1737  ax12olem5  1872  cleqh  2380  hbral  2591  ax12OLD  29105  a12study9  29120  a12study5rev  29122  a12study10  29136  a12study10n  29137
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-11 1715
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