Theorem ax9 1949

Description: Theorem showing that ax-9 1654 follows from the weaker version ax9v 1655. (Even though this theorem depends on ax-9 1654, all references of ax-9 1654 are made via ax9v 1655. An earlier version stated ax9v 1655 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-9 1654 so that all proofs can be traced back to ax9v 1655. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.)

Assertion

Ref	Expression
ax9	|- -. A.x -. x = y

Proof of Theorem ax9

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sp 1747	. . 3 |- (A.x -. x = y -> -. x = y)
2		sp 1747	. . 3 |- (A.x x = y -> x = y)
3	1, 2	nsyl3 111	. 2 |- (A.x x = y -> -. A.x -. x = y)
4		ax9v 1655	. . 3 |- -. A.v -. v = y
5		dveeq2 1940	. . . . . 6 |- (-. A.x x = y -> (v = y -> A.x v = y))
6		ax9v 1655	. . . . . . 7 |- -. A.x -. x = v
7		hba1 1786	. . . . . . . 8 |- (A.x v = y -> A.xA.x v = y)
8		sp 1747	. . . . . . . . . 10 |- (A.x v = y -> v = y)
9		equequ2 1686	. . . . . . . . . 10 |- (v = y -> (x = v <-> x = y))
10	8, 9	syl 15	. . . . . . . . 9 |- (A.x v = y -> (x = v <-> x = y))
11	10	notbid 285	. . . . . . . 8 |- (A.x v = y -> (-. x = v <-> -. x = y))
12	7, 11	albidh 1590	. . . . . . 7 |- (A.x v = y -> (A.x -. x = v <-> A.x -. x = y))
13	6, 12	mtbii 293	. . . . . 6 |- (A.x v = y -> -. A.x -. x = y)
14	5, 13	syl6com 31	. . . . 5 |- (v = y -> (-. A.x x = y -> -. A.x -. x = y))
15	14	con3i 127	. . . 4 |- (-. (-. A.x x = y -> -. A.x -. x = y) -> -. v = y)
16	15	alrimiv 1631	. . 3 |- (-. (-. A.x x = y -> -. A.x -. x = y) -> A.v -. v = y)
17	4, 16	mt3 171	. 2 |- (-. A.x x = y -> -. A.x -. x = y)
18	3, 17	pm2.61i 156	1 |- -. A.x -. x = y

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax9o  1950  a9e  1951