Theorem ax10lem4 1941

Description: Lemma for ax10 1944. Change bound variable. (Contributed by NM, 8-Jul-2016.)

Assertion

Ref	Expression
ax10lem4	|- (A.x x = w -> A.y y = x)

Distinct variable groups:   x,w   y,w

Proof of Theorem ax10lem4

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax10lem1 1936	. . . . . 6 |- (A.x x = w -> A.y y = w)
2		equequ1 1684	. . . . . . . 8 |- (z = x -> (z = w <-> x = w))
3	2	dvelimv 1939	. . . . . . 7 |- (-. A.y y = x -> (x = w -> A.y x = w))
4		hba1 1786	. . . . . . . . 9 |- (A.y x = w -> A.yA.y x = w)
5		equequ2 1686	. . . . . . . . . 10 |- (x = w -> (y = x <-> y = w))
6	5	sps 1754	. . . . . . . . 9 |- (A.y x = w -> (y = x <-> y = w))
7	4, 6	albidh 1590	. . . . . . . 8 |- (A.y x = w -> (A.y y = x <-> A.y y = w))
8	7	biimprd 214	. . . . . . 7 |- (A.y x = w -> (A.y y = w -> A.y y = x))
9	3, 8	syl6 29	. . . . . 6 |- (-. A.y y = x -> (x = w -> (A.y y = w -> A.y y = x)))
10	1, 9	syl7 63	. . . . 5 |- (-. A.y y = x -> (x = w -> (A.x x = w -> A.y y = x)))
11	10	spsd 1755	. . . 4 |- (-. A.y y = x -> (A.x x = w -> (A.x x = w -> A.y y = x)))
12	11	pm2.43d 44	. . 3 |- (-. A.y y = x -> (A.x x = w -> A.y y = x))
13	12	com12 27	. 2 |- (A.x x = w -> (-. A.y y = x -> A.y y = x))
14	13	pm2.18d 103	1 |- (A.x x = w -> A.y y = x)

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:  ax10lem5  1942