Theorem hbimd 1462

Description: Deduction form of bound-variable hypothesis builder hbim 1434. (Contributed by NM, 1-Jan-2002.) (Revised by NM, 2-Feb-2015.)

Hypotheses

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

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.xch))
2	1	imim2d 48	. . 3 |- (ph -> ((ps -> ch) -> (ps -> A.xch)))
3		ax-4 1397	. . . . 5 |- (A.xps -> ps)
4	3	imim1i 54	. . . 4 |- ((ps -> A.xch) -> (A.xps -> A.xch))
5		ax-i5r 1425	. . . 4 |- ((A.xps -> A.xch) -> A.x(A.xps -> ch))
6	4, 5	syl 14	. . 3 |- ((ps -> A.xch) -> A.x(A.xps -> ch))
7	2, 6	syl6 29	. 2 |- (ph -> ((ps -> ch) -> A.x(A.xps -> ch)))
8		hbimd.1	. . 3 |- (ph -> A.xph)
9		hbimd.2	. . . 4 |- (ph -> (ps -> A.xps))
10	9	imim1d 69	. . 3 |- (ph -> ((A.xps -> ch) -> (ps -> ch)))
11	8, 10	alimdh 1353	. 2 |- (ph -> (A.x(A.xps -> ch) -> A.x(ps -> ch)))
12	7, 11	syld 40	1 |- (ph -> ((ps -> ch) -> A.x(ps -> ch)))

Syntax hints:   -> wi 4  A.wal 1240

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-5 1333  ax-gen 1335  ax-4 1397  ax-i5r 1425

This theorem is referenced by:  hbbid  1464  19.21ht  1470  equveli  1639  dvelimfALT2  1695