Theorem ax10 1944

Description: Derive set.mm's original ax-10 2140 from others. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.)

Assertion

Ref	Expression
ax10	|- (A.x x = y -> A.y y = x)

Proof of Theorem ax10

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax9v 1655	. 2 |- -. A.z -. z = x
2		df-ex 1542	. . 3 |- (E.z z = x <-> -. A.z -. z = x)
3		dveeq2 1940	. . . . . . . 8 |- (-. A.y y = x -> (z = x -> A.y z = x))
4	3	imp 418	. . . . . . 7 |- ((-. A.y y = x /\ z = x) -> A.y z = x)
5		ax10lem6 1943	. . . . . . . 8 |- (A.x x = y -> (A.y z = x -> A.x z = x))
6		equcomi 1679	. . . . . . . . 9 |- (z = x -> x = z)
7	6	alimi 1559	. . . . . . . 8 |- (A.x z = x -> A.x x = z)
8	5, 7	syl6 29	. . . . . . 7 |- (A.x x = y -> (A.y z = x -> A.x x = z))
9		ax10lem5 1942	. . . . . . 7 |- (A.x x = z -> A.y y = x)
10	4, 8, 9	syl56 30	. . . . . 6 |- (A.x x = y -> ((-. A.y y = x /\ z = x) -> A.y y = x))
11	10	exp3acom23 1372	. . . . 5 |- (A.x x = y -> (z = x -> (-. A.y y = x -> A.y y = x)))
12		pm2.18 102	. . . . 5 |- ((-. A.y y = x -> A.y y = x) -> A.y y = x)
13	11, 12	syl6 29	. . . 4 |- (A.x x = y -> (z = x -> A.y y = x))
14	13	exlimdv 1636	. . 3 |- (A.x x = y -> (E.z z = x -> A.y y = x))
15	2, 14	syl5bir 209	. 2 |- (A.x x = y -> (-. A.z -. z = x -> A.y y = x))
16	1, 15	mpi 16	1 |- (A.x x = y -> A.y y = x)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  aecom  1946  ax10o  1952  axi10  2331