Theorem hbnt 1667

Description: Closed theorem version of bound-variable hypothesis builder hbn 1668. (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 1524	. . . 4 |- ( A. x ph -> ph )
2	1	con3i 633	. . 3 |- ( -. ph -> -. A. x ph )
3		ax6b 1665	. . 3 |- ( -. A. x ph -> A. x -. A. x ph )
4	2, 3	syl 14	. 2 |- ( -. ph -> A. x -. A. x ph )
5		con3 643	. . 3 |- ( ( ph -> A. x ph ) -> ( -. A. x ph -> -. ph ) )
6	5	al2imi 1472	. 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 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370

This theorem is referenced by:  hbn  1668  hbnd  1669  nfnt  1670