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Theorem basis2 20755

Description: Property of a basis. (Contributed by NM, 17-Jul-2006.)

**Assertion**
Ref	Expression
basis2	$/\$ $/\$ $/\$ $/\$

Proof of Theorem basis2 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isbasis2g 20752	. . . . 5 $/\$
2	1	ibi 256	. . . 4 $/\$
3		ineq1 3807	. . . . . . 7
4		sseq2 3627	. . . . . . . . . 10
5	4	anbi2d 740	. . . . . . . . 9 $/\$ $/\$
6	5	rexbidv 3052	. . . . . . . 8 $/\$ $/\$
7	6	raleqbi1dv 3146	. . . . . . 7 $/\$ $/\$
8	3, 7	syl 17	. . . . . 6 $/\$ $/\$
9		ineq2 3808	. . . . . . 7
10		sseq2 3627	. . . . . . . . . 10
11	10	anbi2d 740	. . . . . . . . 9 $/\$ $/\$
12	11	rexbidv 3052	. . . . . . . 8 $/\$ $/\$
13	12	raleqbi1dv 3146	. . . . . . 7 $/\$ $/\$
14	9, 13	syl 17	. . . . . 6 $/\$ $/\$
15	8, 14	rspc2v 3322	. . . . 5 $/\$ $/\$ $/\$
16		eleq1 2689	. . . . . . . 8
17	16	anbi1d 741	. . . . . . 7 $/\$ $/\$
18	17	rexbidv 3052	. . . . . 6 $/\$ $/\$
19	18	rspccv 3306	. . . . 5 $/\$ $/\$
20	15, 19	syl6com 37	. . . 4 $/\$ $/\$ $/\$
21	2, 20	syl 17	. . 3 $/\$ $/\$
22	21	expd 452	. 2 $/\$
23	22	imp43 621	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

wrex 2913

cin 3573

wss 3574

ctb 20749

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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602

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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-in 3581 df-ss 3588 df-pw 4160 df-uni 4437 df-bases 20750

This theorem is referenced by: tgcl 20773 restbas 20962 txbas 21370 basqtop 21514 tgioo 22599