Theorem hbnd 1643

Description: Deduction form of bound-variable hypothesis builder hbn 1642. (Contributed by NM, 3-Jan-2002.)

Hypotheses

Ref	Expression
hbnd.1	|- ( ph -> A. x ph )
hbnd.2	|- ( ph -> ( ps -> A. x ps ) )

Assertion

Ref	Expression
hbnd	|- ( ph -> ( -. ps -> A. x -. ps ) )

Proof of Theorem hbnd

Step	Hyp	Ref	Expression
1		hbnd.1	. . 3 |- ( ph -> A. x ph )
2		hbnd.2	. . 3 |- ( ph -> ( ps -> A. x ps ) )
3	1, 2	alrimih 1457	. 2 |- ( ph -> A. x ( ps -> A. x ps ) )
4		hbnt 1641	. 2 |- ( A. x ( ps -> A. x ps ) -> ( -. ps -> A. x -. ps ) )
5	3, 4	syl 14	1 |- ( ph -> ( -. ps -> A. x -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349

This theorem is referenced by: (None)