dalem20 - Mathbox for Norm Megill

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Theorem dalem20 33660

Description: Lemma for dath 33703. Show that a second dummy atom

exists outside of the

and

planes (when those planes are equal). (Contributed by NM, 14-Aug-2012.)

**Hypotheses**
Ref	Expression
dalem.ph	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
dalem.l
dalem.j	$.\/$
dalem.a
dalem.ps	$/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
dalem20.o
dalem20.y	$.\/$ $.\/$
dalem20.z	$.\/$ $.\/$

**Assertion**
Ref	Expression
dalem20	$/\$

Distinct variable groups:

$.\/$ ,

Allowed substitution hints:

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

) $.\/$ (

)

(

)

(

)

(

)

**Proof of Theorem dalem20**
Step	Hyp	Ref	Expression
1		dalem.ph	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
2		dalem.l	. . . . 5
3		dalem.j	. . . . 5 $.\/$
4		dalem.a	. . . . 5
5		dalem20.y	. . . . 5 $.\/$ $.\/$
6	1, 2, 3, 4, 5	dalem18 33648	. . . 4
7	6	adantr 465	. . 3 $/\$
8		dalem20.o	. . . . . . 7
9		dalem20.z	. . . . . . 7 $.\/$ $.\/$
10	1, 2, 3, 4, 8, 5, 9	dalem19 33649	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
11	10	ex 434	. . . . 5 $/\$ $/\$ $/\$ $/\$ $.\/$
12	11	ancld 553	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
13	12	reximdva 2932	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$
14	7, 13	mpd 15	. 2 $/\$ $/\$ $/\$ $/\$ $.\/$
15		dalem.ps	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
16		3anass 969	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
17	15, 16	bitri 249	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
18	17	2exbii 1636	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
19		r2ex 2876	. . 3 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
20		r19.42v 2979	. . . 4 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
21	20	rexbii 2858	. . 3 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
22	18, 19, 21	3bitr2ri 274	. 2 $/\$ $/\$ $/\$ $.\/$
23	14, 22	sylib 196	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $/\$ wa 369 $/\$ w3a 965

wceq 1370

wex 1587

wcel 1758

wne 2647

wrex 2799 class class class wbr 4399

cfv 5525 (class class class)co 6199

cbs 14291

cple 14363

cjn 15232

catm 33231

chlt 33318

clpl 33459

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4510 ax-sep 4520 ax-nul 4528 ax-pow 4577 ax-pr 4638 ax-un 6481

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2649 df-ral 2803 df-rex 2804 df-reu 2805 df-rab 2807 df-v 3078 df-sbc 3293 df-csb 3395 df-dif 3438 df-un 3440 df-in 3442 df-ss 3449 df-nul 3745 df-if 3899 df-pw 3969 df-sn 3985 df-pr 3987 df-op 3991 df-uni 4199 df-iun 4280 df-br 4400 df-opab 4458 df-mpt 4459 df-id 4743 df-xp 4953 df-rel 4954 df-cnv 4955 df-co 4956 df-dm 4957 df-rn 4958 df-res 4959 df-ima 4960 df-iota 5488 df-fun 5527 df-fn 5528 df-f 5529 df-f1 5530 df-fo 5531 df-f1o 5532 df-fv 5533 df-riota 6160 df-ov 6202 df-oprab 6203 df-poset 15234 df-plt 15246 df-lub 15262 df-glb 15263 df-join 15264 df-meet 15265 df-p0 15327 df-p1 15328 df-lat 15334 df-clat 15396 df-oposet 33144 df-ol 33146 df-oml 33147 df-covers 33234 df-ats 33235 df-atl 33266 df-cvlat 33290 df-hlat 33319 df-llines 33465 df-lplanes 33466

This theorem is referenced by: dalem62 33701

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