Theorem ax467to7 1373

Description: Re-derivation of ax-7 1304 from ax467 1370. Note that ax-6o 1324 and ax-7 1304 are not used by the re-derivation. The use of alimi 1338 (which uses ax-4 1319) is allowed since we have already proved ax467to4 1371.

Assertion

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

Proof of Theorem ax467to7

Step	Hyp	Ref	Expression
1		ax467to6 1372	. . 3 |- (-. A.y -. A.y -. A.xA.yph -> -. A.xA.yph)
2	1	con4i 90	. 2 |- (A.xA.yph -> A.y -. A.y -. A.xA.yph)
3		pm2.21 92	. . . . . 6 |- (-. A.xA.y -. A.xA.yph -> (A.xA.y -. A.xA.yph -> A.xph))
4		ax467 1370	. . . . . 6 |- ((A.xA.y -. A.xA.yph -> A.xph) -> ph)
5	3, 4	syl 12	. . . . 5 |- (-. A.xA.y -. A.xA.yph -> ph)
6	5	alimi 1338	. . . 4 |- (A.x -. A.xA.y -. A.xA.yph -> A.xph)
7		ax467to6 1372	. . . 4 |- (-. A.x -. A.xA.y -. A.xA.yph -> A.y -. A.xA.yph)
8	6, 7	nsyl4 135	. . 3 |- (-. A.y -. A.xA.yph -> A.xph)
9	8	alimi 1338	. 2 |- (A.y -. A.y -. A.xA.yph -> A.yA.xph)
10	2, 9	syl 12	1 |- (A.xA.yph -> A.yA.xph)

Syntax hints:  -. wn 2   -> wi 3  A.wal 1296

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-4 1319  ax-5o 1321  ax-6o 1324

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