Theorem aceq2 8521

Description: Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. (Contributed by NM, 5-Apr-2004.)

Assertion

Ref	Expression
aceq2

Distinct variable group:   ,,,,,

Proof of Theorem aceq2

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 2812	. . . . 5
2		19.23v 1760	. . . . 5
3	1, 2	bitri 249	. . . 4
4		biidd 237	. . . . 5
5	4	cbvralv 3084	. . . 4
6		n0 3794	. . . . 5
7		eleq2 2530	. . . . . . . . 9
8		eleq2 2530	. . . . . . . . 9
9	7, 8	anbi12d 710	. . . . . . . 8
10	9	cbvrexv 3085	. . . . . . 7
11	10	reubii 3044	. . . . . 6
12		eleq1 2529	. . . . . . . . 9
13	12	anbi2d 703	. . . . . . . 8
14	13	rexbidv 2968	. . . . . . 7
15	14	cbvreuv 3086	. . . . . 6
16	11, 15	bitri 249	. . . . 5
17	6, 16	imbi12i 326	. . . 4
18	3, 5, 17	3bitr4i 277	. . 3
19	18	ralbii 2888	. 2
20	19	exbii 1667	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  E.wex 1612  =/=wne 2652  A.wral 2807  E.wrex 2808  E!wreu 2809  c0 3784

This theorem is referenced by:  dfac7  8533  ac3  8863

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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-v 3111  df-dif 3478  df-nul 3785