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Theorem hbn 1720
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 1716 . . 3  |-  ( A. x ph  ->  ph )
21con3i 127 . 2  |-  ( -. 
ph  ->  -.  A. x ph )
3 hbn1 1704 . . 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 1552 . 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 1527
This theorem is referenced by:  spimeh  1722  hbim  1725  hbex  1733  hban  1736  cbv3hv  1737  hbnae  1895  hbnae-o  2118  vk15.4j  28291  vk15.4jVD  28690  hbnae-x12  29110  equsexv-x12  29113
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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