Theorem zfac 8861

Description: Axiom of Choice expressed with the fewest number of different variables. The penultimate step shows the logical equivalence to ax-ac 8860. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.)

Assertion

Ref	Expression
zfac

Distinct variable group:   ,,,

Proof of Theorem zfac

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-ac 8860	. 2
2		equequ2 1799	. . . . . . . . . 10
3	2	bibi2d 318	. . . . . . . . 9
4		elequ2 1823	. . . . . . . . . . . . 13
5	4	anbi2d 703	. . . . . . . . . . . 12
6		elequ2 1823	. . . . . . . . . . . . 13
7		elequ1 1821	. . . . . . . . . . . . 13
8	6, 7	anbi12d 710	. . . . . . . . . . . 12
9	5, 8	anbi12d 710	. . . . . . . . . . 11
10	9	cbvexv 2024	. . . . . . . . . 10
11	10	bibi1i 314	. . . . . . . . 9
12	3, 11	syl6bb 261	. . . . . . . 8
13	12	albidv 1713	. . . . . . 7
14		elequ1 1821	. . . . . . . . . . . 12
15	14	anbi1d 704	. . . . . . . . . . 11
16		elequ1 1821	. . . . . . . . . . . 12
17	16	anbi1d 704	. . . . . . . . . . 11
18	15, 17	anbi12d 710	. . . . . . . . . 10
19	18	exbidv 1714	. . . . . . . . 9
20		equequ1 1798	. . . . . . . . 9
21	19, 20	bibi12d 321	. . . . . . . 8
22	21	cbvalv 2023	. . . . . . 7
23	13, 22	syl6bb 261	. . . . . 6
24	23	cbvexv 2024	. . . . 5
25	24	imbi2i 312	. . . 4
26	25	2albii 1641	. . 3
27	26	exbii 1667	. 2
28	1, 27	mpbi 208	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612

This theorem is referenced by:  axacndlem4  9009

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ac 8860

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1613  df-nf 1617