Theorem hbimd 1587

Description: Deduction form of bound-variable hypothesis builder hbim 1559. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.)

Hypotheses

Ref	Expression
hbimd.1	|- ( ph -> A. x ph )
hbimd.2	|- ( ph -> ( ps -> A. x ps ) )
hbimd.3	|- ( ph -> ( ch -> A. x ch ) )

Assertion

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

Proof of Theorem hbimd

Step	Hyp	Ref	Expression
1		hbimd.3	. . . 4 |- ( ph -> ( ch -> A. x ch ) )
2	1	imim2d 54	. . 3 |- ( ph -> ( ( ps -> ch ) -> ( ps -> A. x ch ) ) )
3		ax-4 1524	. . . . 5 |- ( A. x ps -> ps )
4	3	imim1i 60	. . . 4 |- ( ( ps -> A. x ch ) -> ( A. x ps -> A. x ch ) )
5		ax-i5r 1549	. . . 4 |- ( ( A. x ps -> A. x ch ) -> A. x ( A. x ps -> ch ) )
6	4, 5	syl 14	. . 3 |- ( ( ps -> A. x ch ) -> A. x ( A. x ps -> ch ) )
7	2, 6	syl6 33	. 2 |- ( ph -> ( ( ps -> ch ) -> A. x ( A. x ps -> ch ) ) )
8		hbimd.1	. . 3 |- ( ph -> A. x ph )
9		hbimd.2	. . . 4 |- ( ph -> ( ps -> A. x ps ) )
10	9	imim1d 75	. . 3 |- ( ph -> ( ( A. x ps -> ch ) -> ( ps -> ch ) ) )
11	8, 10	alimdh 1481	. 2 |- ( ph -> ( A. x ( A. x ps -> ch ) -> A. x ( ps -> ch ) ) )
12	7, 11	syld 45	1 |- ( ph -> ( ( ps -> ch ) -> A. x ( ps -> ch ) ) )

Syntax hints:   -> wi 4   A.wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-5 1461  ax-gen 1463  ax-4 1524  ax-i5r 1549

This theorem is referenced by:  hbbid  1589  19.21ht  1595  equveli  1773  dvelimfALT2  1831