Theorem hbn 2146

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

Step	Hyp	Ref	Expression
1		hbnt 2144	. 2 |- ( A. x ( ph -> A. x ph ) -> ( -. ph -> A. x -. ph ) )
2		hbn.1	. 2 |- ( ph -> A. x ph )
3	1, 2	mpg 1724	1 |- ( -. ph -> A. x -. ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1705  df-nf 1710

This theorem is referenced by:  hbexOLD  2152  hbnae  2317  ac6s6  33980  hbnae-o  34213  vk15.4j  38734  vk15.4jVD  39150