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Theorem basis2 19578
 Description: Property of a basis. (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
basis2
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem basis2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isbasis2g 19575 . . . . 5
21ibi 241 . . . 4
3 ineq1 3689 . . . . . . 7
4 sseq2 3521 . . . . . . . . . 10
54anbi2d 703 . . . . . . . . 9
65rexbidv 2968 . . . . . . . 8
76raleqbi1dv 3062 . . . . . . 7
83, 7syl 16 . . . . . 6
9 ineq2 3690 . . . . . . 7
10 sseq2 3521 . . . . . . . . . 10
1110anbi2d 703 . . . . . . . . 9
1211rexbidv 2968 . . . . . . . 8
1312raleqbi1dv 3062 . . . . . . 7
149, 13syl 16 . . . . . 6
158, 14rspc2v 3219 . . . . 5
16 eleq1 2529 . . . . . . . 8
1716anbi1d 704 . . . . . . 7
1817rexbidv 2968 . . . . . 6
1918rspccv 3207 . . . . 5
2015, 19syl6com 35 . . . 4
212, 20syl 16 . . 3
2221expd 436 . 2
2322imp43 595 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395   wcel 1819  wral 2807  wrex 2808   cin 3470   wss 3471  ctb 19524 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3478  df-ss 3485  df-pw 4017  df-uni 4252  df-bases 19527 This theorem is referenced by:  tgcl  19597  restbas  19785  txbas  20193  basqtop  20337  tgioo  21426
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