Theorem ax6 1020

Description: Rederivation of axiom ax-6 1002 from the orginal version, ax-6o 1019. See ax6o 1018 for the derivation of ax-6o 1019 from ax-6 1002.

This theorem should not be referenced in any proof. Instead, use ax-6 1002 above so that uses of ax-6 1002 can be more easily identified.

Assertion

Ref	Expression
ax6	|- (-. A.xph -> A.x -. A.xph)

Proof of Theorem ax6

Step	Hyp	Ref	Expression
1		ax-4 1014	. . . . 5 |- (A.x -. A.xA.xph -> -. A.xA.xph)
2		id 59	. . . . . . 7 |- (A.xph -> A.xph)
3	2	ax-gen 1004	. . . . . 6 |- A.x(A.xph -> A.xph)
4		ax-5o 1016	. . . . . 6 |- (A.x(A.xph -> A.xph) -> (A.xph -> A.xA.xph))
5	3, 4	ax-mp 7	. . . . 5 |- (A.xph -> A.xA.xph)
6	1, 5	nsyl 122	. . . 4 |- (A.x -. A.xA.xph -> -. A.xph)
7	6	ax-gen 1004	. . 3 |- A.x(A.x -. A.xA.xph -> -. A.xph)
8		ax-5o 1016	. . 3 |- (A.x(A.x -. A.xA.xph -> -. A.xph) -> (A.x -. A.xA.xph -> A.x -. A.xph))
9	7, 8	ax-mp 7	. 2 |- (A.x -. A.xA.xph -> A.x -. A.xph)
10		ax-6o 1019	. 2 |- (-. A.x -. A.xA.xph -> A.xph)
11	9, 10	nsyl4 126	1 |- (-. A.xph -> A.x -. A.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-gen 1004  ax-4 1014  ax-5o 1016  ax-6o 1019

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