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Mirrors > Home > ILE Home > Th. List > ennnfonelemr | Unicode version |
Description: Lemma for ennnfone 12429. The interesting direction, expressed in deduction form. (Contributed by Jim Kingdon, 27-Oct-2022.) |
Ref | Expression |
---|---|
ennnfonelemr.dceq |
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ennnfonelemr.f |
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ennnfonelemr.n |
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Ref | Expression |
---|---|
ennnfonelemr |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ennnfonelemr.dceq |
. . 3
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2 | equequ1 1712 |
. . . . 5
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3 | 2 | dcbid 838 |
. . . 4
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4 | equequ2 1713 |
. . . . 5
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5 | 4 | dcbid 838 |
. . . 4
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6 | 3, 5 | cbvral2v 2718 |
. . 3
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7 | 1, 6 | sylib 122 |
. 2
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8 | ennnfonelemr.f |
. 2
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9 | ennnfonelemr.n |
. . 3
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10 | fveq2 5517 |
. . . . . . . . 9
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11 | 10 | neeq2d 2366 |
. . . . . . . 8
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12 | 11 | cbvralv 2705 |
. . . . . . 7
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13 | 12 | rexbii 2484 |
. . . . . 6
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14 | fveq2 5517 |
. . . . . . . . 9
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15 | 14 | neeq1d 2365 |
. . . . . . . 8
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16 | 15 | ralbidv 2477 |
. . . . . . 7
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17 | 16 | cbvrexv 2706 |
. . . . . 6
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18 | 13, 17 | bitri 184 |
. . . . 5
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19 | 18 | ralbii 2483 |
. . . 4
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20 | oveq2 5886 |
. . . . . . 7
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21 | 20 | raleqdv 2679 |
. . . . . 6
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22 | 21 | rexbidv 2478 |
. . . . 5
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23 | 22 | cbvralv 2705 |
. . . 4
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24 | 19, 23 | bitri 184 |
. . 3
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25 | 9, 24 | sylib 122 |
. 2
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26 | oveq1 5885 |
. . . 4
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27 | 26 | cbvmptv 4101 |
. . 3
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28 | freceq1 6396 |
. . 3
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29 | 27, 28 | ax-mp 5 |
. 2
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30 | 7, 8, 25, 29 | ennnfonelemnn0 12426 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-frec 6395 df-er 6538 df-pm 6654 df-en 6744 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-inn 8923 df-n0 9180 df-z 9257 df-uz 9532 df-fz 10012 df-seqfrec 10449 |
This theorem is referenced by: ennnfone 12429 |
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