Theorem axc711 34199

Description: Proof of a single axiom that can replace both ax-c7 34170 and ax-11 2034. See axc711toc7 34201 and axc711to11 34202 for the rederivation of those axioms. (Contributed by NM, 18-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc711	|- ( -. A. x -. A. y A. x ph -> A. y ph )

Proof of Theorem axc711

Step	Hyp	Ref	Expression
1		ax-11 2034	. . . . 5 |- ( A. y A. x ph -> A. x A. y ph )
2	1	con3i 150	. . . 4 |- ( -. A. x A. y ph -> -. A. y A. x ph )
3	2	alimi 1739	. . 3 |- ( A. x -. A. x A. y ph -> A. x -. A. y A. x ph )
4	3	con3i 150	. 2 |- ( -. A. x -. A. y A. x ph -> -. A. x -. A. x A. y ph )
5		ax-c7 34170	. 2 |- ( -. A. x -. A. x A. y ph -> A. y ph )
6	4, 5	syl 17	1 |- ( -. A. x -. A. y A. x ph -> A. y ph )

Syntax hints:   -. wn 3   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-11 2034  ax-c7 34170

This theorem is referenced by:  axc711toc7  34201  axc711to11  34202