MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbn Structured version   Unicode version

Theorem hbn 1801
Description: If  x is not free in  ph, it is not free in  -.  ph. (Contributed by NM, 5-Aug-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 1799 . 2  |-  ( A. x ( ph  ->  A. x ph )  -> 
( -.  ph  ->  A. x  -.  ph )
)
2 hbn.1 . 2  |-  ( ph  ->  A. x ph )
31, 2mpg 1557 1  |-  ( -. 
ph  ->  A. x  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549
This theorem is referenced by:  hba1  1804  hbimOLD  1837  spimehOLD  1840  hbanOLD  1851  hbex  1863  cbv3hvOLD  1879  hbnae  2043  hbnae-o  2256  vk15.4j  28612  vk15.4jVD  29026  cbv3hvNEW7  29446  hbexwAUX7  29449  hbnaewAUX7  29481  hbnaew2AUX7  29484  ax7w2AUX7  29650  ax7w6AUX7  29652  hbexOLD7  29685  hbnaeOLD7  29710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-ex 1551
  Copyright terms: Public domain W3C validator