Theorem ax467to7 2172

Description: Re-derivation of ax-7 1734 from ax467 2169. Note that ax-6o 2137 and ax-7 1734 are not used by the re-derivation. The use of alimi 1559 (which uses ax-4 2135) is allowed since we have already proved ax467to4 2170. (Contributed by NM, 19-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax467to7	|- (A.xA.yph -> A.yA.xph)

Proof of Theorem ax467to7

Step	Hyp	Ref	Expression
1		ax467to6 2171	. . 3 |- (-. A.y -. A.y -. A.xA.yph -> -. A.xA.yph)
2	1	con4i 122	. 2 |- (A.xA.yph -> A.y -. A.y -. A.xA.yph)
3		pm2.21 100	. . . . . 6 |- (-. A.xA.y -. A.xA.yph -> (A.xA.y -. A.xA.yph -> A.xph))
4		ax467 2169	. . . . . 6 |- ((A.xA.y -. A.xA.yph -> A.xph) -> ph)
5	3, 4	syl 15	. . . . 5 |- (-. A.xA.y -. A.xA.yph -> ph)
6	5	alimi 1559	. . . 4 |- (A.x -. A.xA.y -. A.xA.yph -> A.xph)
7		ax467to6 2171	. . . 4 |- (-. A.x -. A.xA.y -. A.xA.yph -> A.y -. A.xA.yph)
8	6, 7	nsyl4 134	. . 3 |- (-. A.y -. A.xA.yph -> A.xph)
9	8	alimi 1559	. 2 |- (A.y -. A.y -. A.xA.yph -> A.yA.xph)
10	2, 9	syl 15	1 |- (A.xA.yph -> A.yA.xph)

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-7 1734  ax-4 2135  ax-5o 2136  ax-6o 2137

This theorem is referenced by: (None)