4atex2 - Mathbox for Norm Megill

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Theorem 4atex2 36217

Description: More general version of 4atex 36216 for a line

$.\/$

not necessarily connected to

$.\/$

. (Contributed by NM, 27-May-2013.)

**Hypotheses**
Ref	Expression
4that.l
4that.j	$.\/$
4that.a
4that.h

**Assertion**
Ref	Expression
4atex2	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Distinct variable groups:

$.\/$ ,

Allowed substitution hint:

(

)

Proof of Theorem 4atex2 Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6277	. . . . 5 $.\/$ $.\/$
2	1	eqeq1d 2456	. . . 4 $.\/$ $.\/$ $.\/$ $.\/$
3	2	anbi2d 701	. . 3 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
4	3	rexbidv 2965	. 2 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
5		simpl1 997	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$
6		simpl23 1074	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$
7		simpl21 1072	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$
8		simpl32 1076	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
9		simpr 459	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
10		simpl22 1073	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$
11		simp23l 1115	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
12	11	adantr 463	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
13		simpl31 1075	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
14		simpl33 1077	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
15		4that.l	. . . . . 6
16		4that.j	. . . . . 6 $.\/$
17		4that.a	. . . . . 6
18		4that.h	. . . . . 6
19	15, 16, 17, 18	4atex 36216	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
20	5, 7, 10, 12, 13, 14, 19	syl132anc 1244	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
21		eqcom 2463	. . . . . 6 $.\/$ $.\/$ $.\/$ $.\/$
22	21	anbi2i 692	. . . . 5 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
23	22	rexbii 2956	. . . 4 $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
24	20, 23	sylib 196	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
25	15, 16, 17, 18	4atex 36216	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
26	5, 6, 7, 8, 9, 24, 25	syl132anc 1244	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
27		simp1 994	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
28		simp21 1027	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
29		simp22 1028	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$
30		simp32 1031	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
31		simp31 1030	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
32		simp33 1032	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
33	15, 16, 17, 18	4atex 36216	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
34	27, 28, 29, 30, 31, 32, 33	syl132anc 1244	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
35	4, 26, 34	pm2.61ne 2769	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 367 $/\$ w3a 971

wceq 1398

wcel 1823

wne 2649

wrex 2805 class class class wbr 4439

cfv 5570 (class class class)co 6270

cple 14794

cjn 15775

catm 35404

chlt 35491

clh 36124

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1623 ax-4 1636 ax-5 1709 ax-6 1752 ax-7 1795 ax-8 1825 ax-9 1827 ax-10 1842 ax-11 1847 ax-12 1859 ax-13 2004 ax-ext 2432 ax-rep 4550 ax-sep 4560 ax-nul 4568 ax-pow 4615 ax-pr 4676 ax-un 6565

This theorem depends on definitions: df-bi 185 df-or 368 df-an 369 df-3an 973 df-tru 1401 df-ex 1618 df-nf 1622 df-sb 1745 df-eu 2288 df-mo 2289 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2651 df-ral 2809 df-rex 2810 df-reu 2811 df-rab 2813 df-v 3108 df-sbc 3325 df-csb 3421 df-dif 3464 df-un 3466 df-in 3468 df-ss 3475 df-nul 3784 df-if 3930 df-pw 4001 df-sn 4017 df-pr 4019 df-op 4023 df-uni 4236 df-iun 4317 df-br 4440 df-opab 4498 df-mpt 4499 df-id 4784 df-xp 4994 df-rel 4995 df-cnv 4996 df-co 4997 df-dm 4998 df-rn 4999 df-res 5000 df-ima 5001 df-iota 5534 df-fun 5572 df-fn 5573 df-f 5574 df-f1 5575 df-fo 5576 df-f1o 5577 df-fv 5578 df-riota 6232 df-ov 6273 df-oprab 6274 df-preset 15759 df-poset 15777 df-plt 15790 df-lub 15806 df-glb 15807 df-join 15808 df-meet 15809 df-p0 15871 df-p1 15872 df-lat 15878 df-clat 15940 df-oposet 35317 df-ol 35319 df-oml 35320 df-covers 35407 df-ats 35408 df-atl 35439 df-cvlat 35463 df-hlat 35492 df-llines 35638 df-lplanes 35639 df-lhyp 36128

This theorem is referenced by: 4atex3 36221

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