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Theorem isfildlem 20950
 Description: Lemma for isfild 20951. (Contributed by Mario Carneiro, 1-Dec-2013.)
Hypotheses
Ref Expression
isfild.1
isfild.2
Assertion
Ref Expression
isfildlem
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem isfildlem
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3040 . . 3
21a1i 11 . 2
3 isfild.2 . . . 4
4 ssexg 4542 . . . . 5
54expcom 442 . . . 4
63, 5syl 17 . . 3
8 eleq1 2537 . . . . . 6
9 sseq1 3439 . . . . . . 7
10 dfsbcq 3257 . . . . . . 7
119, 10anbi12d 725 . . . . . 6
128, 11bibi12d 328 . . . . 5
1312imbi2d 323 . . . 4
14 nfv 1769 . . . . . 6
15 nfv 1769 . . . . . . 7
16 nfv 1769 . . . . . . . 8
17 nfsbc1v 3275 . . . . . . . 8
1816, 17nfan 2031 . . . . . . 7
1915, 18nfbi 2037 . . . . . 6
2014, 19nfim 2023 . . . . 5
21 eleq1 2537 . . . . . . 7
22 sseq1 3439 . . . . . . . 8
23 sbceq1a 3266 . . . . . . . 8
2422, 23anbi12d 725 . . . . . . 7
2521, 24bibi12d 328 . . . . . 6
2625imbi2d 323 . . . . 5
27 isfild.1 . . . . 5
2820, 26, 27chvar 2119 . . . 4
2913, 28vtoclg 3093 . . 3
3029com12 31 . 2
312, 7, 30pm5.21ndd 361 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   wceq 1452   wcel 1904  cvv 3031  wsbc 3255   wss 3390 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518 This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-sbc 3256  df-in 3397  df-ss 3404 This theorem is referenced by:  isfild  20951
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