dalem20 - Mathbox for Norm Megill

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Theorem dalem20 33303

Description: Lemma for dath 33346. 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 33291	. . . 4
7	6	adantr 471	. . 3 $/\$
8		dalem20.o	. . . . . . 7
9		dalem20.z	. . . . . . 7 $.\/$ $.\/$
10	1, 2, 3, 4, 8, 5, 9	dalem19 33292	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
11	10	ex 440	. . . . 5 $/\$ $/\$ $/\$ $/\$ $.\/$
12	11	ancld 560	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
13	12	reximdva 2874	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$
14	7, 13	mpd 15	. 2 $/\$ $/\$ $/\$ $/\$ $.\/$
15		dalem.ps	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
16		3anass 995	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
17	15, 16	bitri 257	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
18	17	2exbii 1730	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
19		r2ex 2925	. . 3 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
20		r19.42v 2957	. . . 4 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
21	20	rexbii 2901	. . 3 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
22	18, 19, 21	3bitr2ri 282	. 2 $/\$ $/\$ $/\$ $.\/$
23	14, 22	sylib 201	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 189 $/\$ wa 375 $/\$ w3a 991

wceq 1455

wex 1674

wcel 1898

wne 2633

wrex 2750 class class class wbr 4416

cfv 5601 (class class class)co 6315

cbs 15170

cple 15246

cjn 16238

catm 32874

chlt 32961

clpl 33102

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4529 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610

This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-iun 4294 df-br 4417 df-opab 4476 df-mpt 4477 df-id 4768 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-riota 6277 df-ov 6318 df-oprab 6319 df-preset 16222 df-poset 16240 df-plt 16253 df-lub 16269 df-glb 16270 df-join 16271 df-meet 16272 df-p0 16334 df-p1 16335 df-lat 16341 df-clat 16403 df-oposet 32787 df-ol 32789 df-oml 32790 df-covers 32877 df-ats 32878 df-atl 32909 df-cvlat 32933 df-hlat 32962 df-llines 33108 df-lplanes 33109

This theorem is referenced by: dalem62 33344

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