Theorem kmlem16 8566

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4 5 <=> 4. (Contributed by NM, 4-Apr-2004.)

Hypotheses

Ref	Expression
kmlem14.1
kmlem14.2
kmlem14.3

Assertion

Ref	Expression
kmlem16

Distinct variable groups:   ,,,,,   ,

Proof of Theorem kmlem16

Step	Hyp	Ref	Expression
1		kmlem14.1	. . . 4
2		kmlem14.2	. . . 4
3		kmlem14.3	. . . 4
4	1, 2, 3	kmlem14 8564	. . 3
5	1, 2, 3	kmlem15 8565	. . . 4
6	5	exbii 1667	. . 3
7	4, 6	orbi12i 521	. 2
8		19.43 1693	. 2
9		pm3.24 882	. . . . . 6
10		simpl 457	. . . . . . . . 9
11	10	sps 1865	. . . . . . . 8
12	11	exlimivv 1723	. . . . . . 7
13		simpl 457	. . . . . . . . 9
14	13	sps 1865	. . . . . . . 8
15	14	exlimivv 1723	. . . . . . 7
16	12, 15	anim12i 566	. . . . . 6
17	9, 16	mto 176	. . . . 5
18		19.33b 1696	. . . . 5
19	17, 18	ax-mp 5	. . . 4
20	10	exlimiv 1722	. . . . . . . . . 10
21	13	exlimiv 1722	. . . . . . . . . 10
22	20, 21	anim12i 566	. . . . . . . . 9
23	9, 22	mto 176	. . . . . . . 8
24		19.33b 1696	. . . . . . . 8
25	23, 24	ax-mp 5	. . . . . . 7
26	25	exbii 1667	. . . . . 6
27		19.43 1693	. . . . . 6
28	26, 27	bitr2i 250	. . . . 5
29	28	albii 1640	. . . 4
30	19, 29	bitr3i 251	. . 3
31	30	exbii 1667	. 2
32	7, 8, 31	3bitr2i 273	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  A.wal 1393  E.wex 1612  e.wcel 1818  E!weu 2282  =/=wne 2652  A.wral 2807  E.wrex 2808  i^icin 3474

This theorem is referenced by:  dfackm  8567

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-v 3111  df-in 3482