Theorem ax467to7 1067

Description: Re-derivation of ax-7 1003 from ax467 1064. Note that ax-6o 1019 and ax-7 1003 are not used by the re-derivation. The use of 19.20i 1033 (which uses ax-4 1014) is allowed since we have already proved ax467to4 1065.

Assertion

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

Proof of Theorem ax467to7

Step	Hyp	Ref	Expression
1		ax467to6 1066	. . 3 |- (-. A.y -. A.y -. A.xA.yph -> -. A.xA.yph)
2	1	a3i 77	. 2 |- (A.xA.yph -> A.y -. A.y -. A.xA.yph)
3		pm2.21 79	. . . . . 6 |- (-. A.xA.y -. A.xA.yph -> (A.xA.y -. A.xA.yph -> A.xph))
4		ax467 1064	. . . . . 6 |- ((A.xA.y -. A.xA.yph -> A.xph) -> ph)
5	3, 4	syl 10	. . . . 5 |- (-. A.xA.y -. A.xA.yph -> ph)
6	5	19.20i 1033	. . . 4 |- (A.x -. A.xA.y -. A.xA.yph -> A.xph)
7		ax467to6 1066	. . . 4 |- (-. A.x -. A.xA.y -. A.xA.yph -> A.y -. A.xA.yph)
8	6, 7	nsyl4 126	. . 3 |- (-. A.y -. A.xA.yph -> A.xph)
9	8	19.20i 1033	. 2 |- (A.y -. A.y -. A.xA.yph -> A.yA.xph)
10	2, 9	syl 10	1 |- (A.xA.yph -> A.yA.xph)

Syntax hints:  -. wn 2   -> wi 3  A.wal 995

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-4 1014  ax-5o 1016  ax-6o 1019

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