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Theorem hbn 1834
Description: If  x is not free in  ph, it is not free in  -.  ph. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 17-Dec-2017.)
Hypothesis
Ref Expression
hbn.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
hbn  |-  ( -. 
ph  ->  A. x  -.  ph )

Proof of Theorem hbn
StepHypRef Expression
1 hbnt 1833 . 2  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  ->  A. x  -.  ph )
)
2 hbn.1 . 2  |-  ( ph  ->  A. x ph )
31, 2mpg 1594 1  |-  ( -. 
ph  ->  A. x  -.  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794
This theorem depends on definitions:  df-bi 185  df-ex 1588
This theorem is referenced by:  hba1  1835  hbex  1884  hbnae  2017  hbnae-o  2238  ac6s6  29133  vk15.4j  31566  vk15.4jVD  31983
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