Theorem ax467 997

Description: Proof of a single axiom that can replace ax-4 951, ax-6 953, and ax-7 954. See ax467to4 998, ax467to6 999, and ax467to7 1000 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 991.

Assertion

Ref	Expression
ax467	|- ((A.xA.y -. A.xA.yph -> A.xph) -> ph)

Proof of Theorem ax467

Step	Hyp	Ref	Expression
1		ax-4 951	. . 3 |- (A.yph -> ph)
2		hbn1 989	. . . 4 |- (-. A.yph -> A.y -. A.yph)
3		ax-6 953	. . . . . 6 |- (-. A.x -. A.xA.yph -> A.yph)
4	3	con1i 96	. . . . 5 |- (-. A.yph -> A.x -. A.xA.yph)
5	4	19.20i 968	. . . 4 |- (A.y -. A.yph -> A.yA.x -. A.xA.yph)
6		ax-7 954	. . . 4 |- (A.yA.x -. A.xA.yph -> A.xA.y -. A.xA.yph)
7	2, 5, 6	3syl 20	. . 3 |- (-. A.yph -> A.xA.y -. A.xA.yph)
8	1, 7	nsyl4 120	. 2 |- (-. A.xA.y -. A.xA.yph -> ph)
9		ax-4 951	. 2 |- (A.xph -> ph)
10	8, 9	ja 137	1 |- ((A.xA.y -. A.xA.yph -> A.xph) -> ph)

Syntax hints:  -. wn 2   -> wi 3  A.wal 950

This theorem is referenced by:  ax467to4 998  ax467to6 999  ax467to7 1000

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955

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