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Theorem wfisg 5422
 Description: Well-Founded Induction Schema. If a property passes from all elements less than of a well-founded class to itself (induction hypothesis), then the property holds for all elements of . (Contributed by Scott Fenton, 11-Feb-2011.)
Hypothesis
Ref Expression
wfisg.1
Assertion
Ref Expression
wfisg Se
Distinct variable groups:   ,,   ,   ,,
Allowed substitution hint:   ()

Proof of Theorem wfisg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssrab2 3500 . . 3
2 dfss3 3408 . . . . . . . 8
3 nfcv 2612 . . . . . . . . . . 11
43elrabsf 3294 . . . . . . . . . 10
54simprbi 471 . . . . . . . . 9
65ralimi 2796 . . . . . . . 8
72, 6sylbi 200 . . . . . . 7
8 nfv 1769 . . . . . . . . 9
9 nfcv 2612 . . . . . . . . . . 11
10 nfsbc1v 3275 . . . . . . . . . . 11
119, 10nfral 2789 . . . . . . . . . 10
12 nfsbc1v 3275 . . . . . . . . . 10
1311, 12nfim 2023 . . . . . . . . 9
148, 13nfim 2023 . . . . . . . 8
15 eleq1 2537 . . . . . . . . 9
16 predeq3 5391 . . . . . . . . . . 11
1716raleqdv 2979 . . . . . . . . . 10
18 sbceq1a 3266 . . . . . . . . . 10
1917, 18imbi12d 327 . . . . . . . . 9
2015, 19imbi12d 327 . . . . . . . 8
21 wfisg.1 . . . . . . . 8
2214, 20, 21chvar 2119 . . . . . . 7
237, 22syl5 32 . . . . . 6
2423anc2li 566 . . . . 5
253elrabsf 3294 . . . . 5
2624, 25syl6ibr 235 . . . 4
2726rgen 2766 . . 3
28 wfi 5420 . . 3 Se
291, 27, 28mpanr12 699 . 2 Se
30 rabid2 2954 . 2
3129, 30sylib 201 1 Se
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wceq 1452   wcel 1904  wral 2756  crab 2760  wsbc 3255   wss 3390   Se wse 4796   wwe 4797  cpred 5386 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-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387 This theorem is referenced by:  wfis  5423  wfis2fg  5424
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