Theorem cbv3hvOLD 1851

Description: Obsolete proof of cbv3hv 1850 as of 29-Dec-2017. (Contributed by NM, 25-Jul-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cbv3hv.1	|- (ph -> A.yph)
cbv3hv.2	|- (ps -> A.xps)
cbv3hv.3	|- (x = y -> (ph -> ps))

Assertion

Ref	Expression
cbv3hvOLD	|- (A.xph -> A.yps)

Distinct variable group:   x,y

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

Proof of Theorem cbv3hvOLD

Step	Hyp	Ref	Expression
1		cbv3hv.1	. . 3 |- (ph -> A.yph)
2	1	alimi 1559	. 2 |- (A.xph -> A.xA.yph)
3		ax9v 1655	. . . . 5 |- -. A.x -. x = y
4		hba1 1786	. . . . . . . 8 |- (A.xph -> A.xA.xph)
5		cbv3hv.2	. . . . . . . 8 |- (ps -> A.xps)
6	4, 5	hbim 1817	. . . . . . 7 |- ((A.xph -> ps) -> A.x(A.xph -> ps))
7	6	hbn 1776	. . . . . 6 |- (-. (A.xph -> ps) -> A.x -. (A.xph -> ps))
8		sp 1747	. . . . . . . 8 |- (A.xph -> ph)
9		cbv3hv.3	. . . . . . . 8 |- (x = y -> (ph -> ps))
10	8, 9	syl5 28	. . . . . . 7 |- (x = y -> (A.xph -> ps))
11	10	con3i 127	. . . . . 6 |- (-. (A.xph -> ps) -> -. x = y)
12	7, 11	alrimih 1565	. . . . 5 |- (-. (A.xph -> ps) -> A.x -. x = y)
13	3, 12	mt3 171	. . . 4 |- (A.xph -> ps)
14	13	alimi 1559	. . 3 |- (A.yA.xph -> A.yps)
15	14	a7s 1735	. 2 |- (A.xA.yph -> A.yps)
16	2, 15	syl 15	1 |- (A.xph -> A.yps)

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

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-7 1734  ax-11 1746

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

This theorem is referenced by: (None)