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Theorem isfildlem 17881
 Description: Lemma for isfild 17882. (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 2956 . . 3
21a1i 11 . 2
3 isfild.2 . . . 4
4 ssexg 4341 . . . . 5
54expcom 425 . . . 4
63, 5syl 16 . . 3
76adantrd 455 . 2
8 eleq1 2495 . . . . . 6
9 sseq1 3361 . . . . . . 7
10 dfsbcq 3155 . . . . . . 7
119, 10anbi12d 692 . . . . . 6
128, 11bibi12d 313 . . . . 5
1312imbi2d 308 . . . 4
14 nfv 1629 . . . . . 6
15 nfv 1629 . . . . . . 7
16 nfv 1629 . . . . . . . 8
17 nfsbc1v 3172 . . . . . . . 8
1816, 17nfan 1846 . . . . . . 7
1915, 18nfbi 1856 . . . . . 6
2014, 19nfim 1832 . . . . 5
21 eleq1 2495 . . . . . . 7
22 sseq1 3361 . . . . . . . 8
23 sbceq1a 3163 . . . . . . . 8
2422, 23anbi12d 692 . . . . . . 7
2521, 24bibi12d 313 . . . . . 6
2625imbi2d 308 . . . . 5
27 isfild.1 . . . . 5
2820, 26, 27chvar 1968 . . . 4
2913, 28vtoclg 3003 . . 3
3029com12 29 . 2
312, 7, 30pm5.21ndd 344 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cvv 2948  wsbc 3153   wss 3312 This theorem is referenced by:  isfild  17882 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154  df-in 3319  df-ss 3326
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