Theorem hbnt 2144

Description: Closed theorem version of bound-variable hypothesis builder hbn 2146. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) (Proof shortened by Wolf Lammen, 14-Oct-2021.)

Assertion

Ref	Expression
hbnt	|- ( A. x ( ph -> A. x ph ) -> ( -. ph -> A. x -. ph ) )

Proof of Theorem hbnt

Step	Hyp	Ref	Expression
1		nf5-1 2023	. . 3 |- ( A. x ( ph -> A. x ph ) -> F/ x ph )
2	1	nfnd 1785	. 2 |- ( A. x ( ph -> A. x ph ) -> F/ x -. ph )
3	2	nf5rd 2066	1 |- ( A. x ( ph -> A. x ph ) -> ( -. 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:  hbn  2146  hbnd  2147  nfntOLD  2209  bj-hbext  32701