Theorem ax67 1061

Description: Proof of a single axiom that can replace both ax-6o 1019 and ax-7 1003. See ax67to6 1062 and ax67to7 1063 for the re-derivation of those axioms.

Assertion

Ref	Expression
ax67	|- (-. A.x -. A.yA.xph -> A.yph)

Proof of Theorem ax67

Step	Hyp	Ref	Expression
1		ax-7 1003	. . . . 5 |- (A.yA.xph -> A.xA.yph)
2	1	con3i 104	. . . 4 |- (-. A.xA.yph -> -. A.yA.xph)
3	2	19.20i 1033	. . 3 |- (A.x -. A.xA.yph -> A.x -. A.yA.xph)
4	3	con3i 104	. 2 |- (-. A.x -. A.yA.xph -> -. A.x -. A.xA.yph)
5		ax-6o 1019	. 2 |- (-. A.x -. A.xA.yph -> A.yph)
6	4, 5	syl 10	1 |- (-. A.x -. A.yA.xph -> A.yph)

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

This theorem is referenced by:  ax67to6 1062  ax67to7 1063

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

Copyright terms: Public domain