Theorem ax10-16 2190

Description: This theorem shows that, given ax-16 2144, we can derive a version of ax-10 2140. However, it is weaker than ax-10 2140 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ax10-16	|- (A.x x = z -> A.z z = x)

Distinct variable group:   x,z

Proof of Theorem ax10-16

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-16 2144	. . . 4 |- (A.x x = z -> (x = w -> A.x x = w))
2	1	alrimiv 1631	. . 3 |- (A.x x = z -> A.w(x = w -> A.x x = w))
3	2	a5i-o 2150	. 2 |- (A.x x = z -> A.xA.w(x = w -> A.x x = w))
4		equequ1 1684	. . . . . 6 |- (x = z -> (x = w <-> z = w))
5	4	cbvalv 2002	. . . . . . 7 |- (A.x x = w <-> A.z z = w)
6	5	a1i 10	. . . . . 6 |- (x = z -> (A.x x = w <-> A.z z = w))
7	4, 6	imbi12d 311	. . . . 5 |- (x = z -> ((x = w -> A.x x = w) <-> (z = w -> A.z z = w)))
8	7	albidv 1625	. . . 4 |- (x = z -> (A.w(x = w -> A.x x = w) <-> A.w(z = w -> A.z z = w)))
9	8	cbvalv 2002	. . 3 |- (A.xA.w(x = w -> A.x x = w) <-> A.zA.w(z = w -> A.z z = w))
10	9	biimpi 186	. 2 |- (A.xA.w(x = w -> A.x x = w) -> A.zA.w(z = w -> A.z z = w))
11		nfa1-o 2166	. . . . . . 7 |- F/zA.z z = w
12	11	19.23 1801	. . . . . 6 |- (A.z(z = w -> A.z z = w) <-> (E.z z = w -> A.z z = w))
13	12	albii 1566	. . . . 5 |- (A.wA.z(z = w -> A.z z = w) <-> A.w(E.z z = w -> A.z z = w))
14		a9ev 1656	. . . . . . . 8 |- E.z z = w
15		pm2.27 35	. . . . . . . 8 |- (E.z z = w -> ((E.z z = w -> A.z z = w) -> A.z z = w))
16	14, 15	ax-mp 5	. . . . . . 7 |- ((E.z z = w -> A.z z = w) -> A.z z = w)
17	16	alimi 1559	. . . . . 6 |- (A.w(E.z z = w -> A.z z = w) -> A.wA.z z = w)
18		equequ2 1686	. . . . . . . . 9 |- (w = x -> (z = w <-> z = x))
19	18	spv 1998	. . . . . . . 8 |- (A.w z = w -> z = x)
20	19	sps-o 2159	. . . . . . 7 |- (A.zA.w z = w -> z = x)
21	20	a7s 1735	. . . . . 6 |- (A.wA.z z = w -> z = x)
22	17, 21	syl 15	. . . . 5 |- (A.w(E.z z = w -> A.z z = w) -> z = x)
23	13, 22	sylbi 187	. . . 4 |- (A.wA.z(z = w -> A.z z = w) -> z = x)
24	23	a7s 1735	. . 3 |- (A.zA.w(z = w -> A.z z = w) -> z = x)
25	24	a5i-o 2150	. 2 |- (A.zA.w(z = w -> A.z z = w) -> A.z z = x)
26	3, 10, 25	3syl 18	1 |- (A.x x = z -> A.z z = x)

Syntax hints:   -> wi 4   <-> wb 176  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-7 1734  ax-11 1746  ax-12 1925  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-16 2144

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: (None)