kmlem8 - Metamath Proof Explorer

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Theorem kmlem8 8979

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4 1 <=> 4. (Contributed by NM, 4-Apr-2004.)

**Assertion**
Ref	Expression
kmlem8	$\/$ $/\$

(

)

**Proof of Theorem kmlem8**
Step	Hyp	Ref	Expression
1		ralnex 2992	. . . . 5
2		df-rex 2918	. . . . . . . 8 $/\$
3		rexnal 2995	. . . . . . . 8
4	2, 3	bitr3i 266	. . . . . . 7 $/\$
5		exsimpl 1795	. . . . . . . 8 $/\$
6		n0 3931	. . . . . . . 8
7	5, 6	sylibr 224	. . . . . . 7 $/\$
8	4, 7	sylbir 225	. . . . . 6
9	8	ralimi 2952	. . . . 5
10	1, 9	sylbir 225	. . . 4
11		biimt 350	. . . . . . . . 9
12	11	ralimi 2952	. . . . . . . 8
13		ralbi 3068	. . . . . . . 8
14	12, 13	syl 17	. . . . . . 7
15	14	anbi2d 740	. . . . . 6 $/\$ $/\$
16	15	exbidv 1850	. . . . 5 $/\$ $/\$
17		kmlem2 8973	. . . . 5 $/\$
18	16, 17	syl6rbbr 279	. . . 4 $/\$
19	10, 18	syl 17	. . 3 $/\$
20	19	pm5.74i 260	. 2 $/\$
21		pm4.64 387	. 2 $/\$ $\/$ $/\$
22	20, 21	bitri 264	1 $\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384

wex 1704

wcel 1990

weu 2470

wne 2794

wral 2912

wrex 2913

cin 3573

c0 3915

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-sn 4178 df-pr 4180 df-uni 4437

This theorem is referenced by: dfackm 8988