Theorem ax467 1370

Description: Proof of a single axiom that can replace ax-4 1319, ax-6o 1324, and ax-7 1304 in a subsystem that includes these axioms plus ax-5o 1321 and ax-gen 1305 (and propositional calculus). See ax467to4 1371, ax467to6 1372, and ax467to7 1373 for the re-derivation of those axioms. This theorem extends the idea in Scott Fenton's ax46 1364.

Assertion

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

Proof of Theorem ax467

Step	Hyp	Ref	Expression
1		ax-4 1319	. . 3 |- (A.yph -> ph)
2		hbn1 1362	. . . 4 |- (-. A.yph -> A.y -. A.yph)
3		ax-6o 1324	. . . . . 6 |- (-. A.x -. A.xA.yph -> A.yph)
4	3	con1i 112	. . . . 5 |- (-. A.yph -> A.x -. A.xA.yph)
5	4	alimi 1338	. . . 4 |- (A.y -. A.yph -> A.yA.x -. A.xA.yph)
6		ax-7 1304	. . . 4 |- (A.yA.x -. A.xA.yph -> A.xA.y -. A.xA.yph)
7	2, 5, 6	3syl 24	. . 3 |- (-. A.yph -> A.xA.y -. A.xA.yph)
8	1, 7	nsyl4 135	. 2 |- (-. A.xA.y -. A.xA.yph -> ph)
9		ax-4 1319	. 2 |- (A.xph -> ph)
10	8, 9	ja 152	1 |- ((A.xA.y -. A.xA.yph -> A.xph) -> ph)

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

This theorem is referenced by:  ax467to4 1371  ax467to6 1372  ax467to7 1373

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

Copyright terms: Public domain