Theorem ax10lem2 1937

Description: Lemma for ax10 1944. Change free variable. (Contributed by NM, 25-Jul-2015.)

Assertion

Ref	Expression
ax10lem2	|- (A.x x = y -> A.x x = z)

Distinct variable groups:   x,y   x,z

Proof of Theorem ax10lem2

Step	Hyp	Ref	Expression
1		hbe1 1731	. . . 4 |- (E.x -. x = y -> A.xE.x -. x = y)
2		equequ2 1686	. . . . . . . 8 |- (z = y -> (x = z <-> x = y))
3	2	biimprd 214	. . . . . . 7 |- (z = y -> (x = y -> x = z))
4	3	con3rr3 128	. . . . . 6 |- (-. x = z -> (z = y -> -. x = y))
5		19.8a 1756	. . . . . 6 |- (-. x = y -> E.x -. x = y)
6	4, 5	syl6 29	. . . . 5 |- (-. x = z -> (z = y -> E.x -. x = y))
7		ax-17 1616	. . . . . 6 |- (-. z = y -> A.x -. z = y)
8		equequ1 1684	. . . . . . . 8 |- (x = z -> (x = y <-> z = y))
9	8	notbid 285	. . . . . . 7 |- (x = z -> (-. x = y <-> -. z = y))
10	9	biimprd 214	. . . . . 6 |- (x = z -> (-. z = y -> -. x = y))
11	7, 10	spimeh 1667	. . . . 5 |- (-. z = y -> E.x -. x = y)
12	6, 11	pm2.61d1 151	. . . 4 |- (-. x = z -> E.x -. x = y)
13	1, 12	exlimih 1804	. . 3 |- (E.x -. x = z -> E.x -. x = y)
14		exnal 1574	. . 3 |- (E.x -. x = z <-> -. A.x x = z)
15		exnal 1574	. . 3 |- (E.x -. x = y <-> -. A.x x = y)
16	13, 14, 15	3imtr3i 256	. 2 |- (-. A.x x = z -> -. A.x x = y)
17	16	con4i 122	1 |- (A.x x = y -> A.x x = z)

Syntax hints:  -. wn 3   -> wi 4  A.wal 1540  E.wex 1541

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-11 1746

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

This theorem is referenced by:  ax10lem3  1938