Theorem hbnt 1583

Description: Closed theorem version of bound-variable hypothesis builder hbn 1584. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

Assertion

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

Proof of Theorem hbnt

Step	Hyp	Ref	Expression
1		ax-4 1440	. . . 4 |- ( A. x ph -> ph )
2	1	con3i 594	. . 3 |- ( -. ph -> -. A. x ph )
3		ax6b 1581	. . 3 |- ( -. A. x ph -> A. x -. A. x ph )
4	2, 3	syl 14	. 2 |- ( -. ph -> A. x -. A. x ph )
5		con3 603	. . 3 |- ( ( ph -> A. x ph ) -> ( -. A. x ph -> -. ph ) )
6	5	al2imi 1387	. 2 |- ( A. x ( ph -> A. x ph ) -> ( A. x -. A. x ph -> A. x -. ph ) )
7	4, 6	syl5 32	1 |- ( A. x ( ph -> A. x ph ) -> ( -. ph -> A. x -. ph ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290

This theorem is referenced by:  hbn  1584  hbnd  1585  nfnt  1586