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Theorem hbn 1732
Description: If  x is not free in  ph, it is not free in  -.  ph. (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
hbn.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
hbn  |-  ( -. 
ph  ->  A. x  -.  ph )

Proof of Theorem hbn
StepHypRef Expression
1 sp 1728 . . 3  |-  ( A. x ph  ->  ph )
21con3i 127 . 2  |-  ( -. 
ph  ->  -.  A. x ph )
3 hbn1 1716 . . 3  |-  ( -. 
A. x ph  ->  A. x  -.  A. x ph )
4 hbn.1 . . . 4  |-  ( ph  ->  A. x ph )
54con3i 127 . . 3  |-  ( -. 
A. x ph  ->  -. 
ph )
63, 5alrimih 1555 . 2  |-  ( -. 
A. x ph  ->  A. x  -.  ph )
72, 6syl 15 1  |-  ( -. 
ph  ->  A. x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530
This theorem is referenced by:  spimeh  1734  hbim  1737  hbex  1745  hban  1748  cbv3hv  1749  hbnae  1908  hbnae-o  2131  vk15.4j  28590  vk15.4jVD  29006  cbv3hvNEW7  29423  hbexwAUX7  29426  hbnaewAUX7  29458  hbnaew2AUX7  29461  ax7w2AUX7  29620  ax7w6AUX7  29622  hbexOLD7  29639  hbnaeOLD7  29664  hbnae-x12  29732  equsexv-x12  29735
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-11 1727
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