Theorem spimehOLD 1821

Description: Obsolete proof of spimeh 1667 as of 10-Dec-2017. (Contributed by NM, 7-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
spimehOLD.1	|- (ph -> A.xph)
spimehOLD.2	|- (x = z -> (ph -> ps))

Assertion

Ref	Expression
spimehOLD	|- (ph -> E.xps)

Distinct variable group:   x,z

Allowed substitution hints:   ph(x,z)   ps(x,z)

Proof of Theorem spimehOLD

Step	Hyp	Ref	Expression
1		ax9v 1655	. . . 4 |- -. A.x -. x = z
2		id 19	. . . . . . 7 |- (ph -> ph)
3	2	hbth 1552	. . . . . . . 8 |- ((ph -> ph) -> A.x(ph -> ph))
4		hba1 1786	. . . . . . . . 9 |- (A.x -. ps -> A.xA.x -. ps)
5	4	a1i 10	. . . . . . . 8 |- ((ph -> ph) -> (A.x -. ps -> A.xA.x -. ps))
6		spimehOLD.1	. . . . . . . . . 10 |- (ph -> A.xph)
7	6	hbn 1776	. . . . . . . . 9 |- (-. ph -> A.x -. ph)
8	7	a1i 10	. . . . . . . 8 |- ((ph -> ph) -> (-. ph -> A.x -. ph))
9	3, 5, 8	hbimd 1815	. . . . . . 7 |- ((ph -> ph) -> ((A.x -. ps -> -. ph) -> A.x(A.x -. ps -> -. ph)))
10	2, 9	ax-mp 5	. . . . . 6 |- ((A.x -. ps -> -. ph) -> A.x(A.x -. ps -> -. ph))
11	10	hbn 1776	. . . . 5 |- (-. (A.x -. ps -> -. ph) -> A.x -. (A.x -. ps -> -. ph))
12		spimehOLD.2	. . . . . . 7 |- (x = z -> (ph -> ps))
13		sp 1747	. . . . . . 7 |- (A.x -. ps -> -. ps)
14	12, 13	nsyli 133	. . . . . 6 |- (x = z -> (A.x -. ps -> -. ph))
15	14	con3i 127	. . . . 5 |- (-. (A.x -. ps -> -. ph) -> -. x = z)
16	11, 15	alrimih 1565	. . . 4 |- (-. (A.x -. ps -> -. ph) -> A.x -. x = z)
17	1, 16	mt3 171	. . 3 |- (A.x -. ps -> -. ph)
18	17	con2i 112	. 2 |- (ph -> -. A.x -. ps)
19		df-ex 1542	. 2 |- (E.xps <-> -. A.x -. ps)
20	18, 19	sylibr 203	1 |- (ph -> E.xps)

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by: (None)