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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-findis | Unicode version |
Description: Principle of induction, using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See bj-bdfindis 13134 for a bounded version not requiring ax-setind 4447. See finds 4509 for a proof in IZF. From this version, it is easy to prove of finds 4509, finds2 4510, finds1 4511. (Contributed by BJ, 22-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-findis.nf0 | |
bj-findis.nf1 | |
bj-findis.nfsuc | |
bj-findis.0 | |
bj-findis.1 | |
bj-findis.suc |
Ref | Expression |
---|---|
bj-findis |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nn0suc 13151 | . . . . 5 | |
2 | pm3.21 262 | . . . . . . . 8 | |
3 | 2 | ad2antrr 479 | . . . . . . 7 |
4 | pm2.04 82 | . . . . . . . . . . 11 | |
5 | 4 | ralimi2 2490 | . . . . . . . . . 10 |
6 | imim2 55 | . . . . . . . . . . . 12 | |
7 | 6 | ral2imi 2495 | . . . . . . . . . . 11 |
8 | 7 | imp 123 | . . . . . . . . . 10 |
9 | 5, 8 | sylan2 284 | . . . . . . . . 9 |
10 | r19.29 2567 | . . . . . . . . . . 11 | |
11 | vex 2684 | . . . . . . . . . . . . . . . 16 | |
12 | 11 | sucid 4334 | . . . . . . . . . . . . . . 15 |
13 | eleq2 2201 | . . . . . . . . . . . . . . 15 | |
14 | 12, 13 | mpbiri 167 | . . . . . . . . . . . . . 14 |
15 | ax-1 6 | . . . . . . . . . . . . . . 15 | |
16 | pm2.27 40 | . . . . . . . . . . . . . . 15 | |
17 | 15, 16 | anim12ii 340 | . . . . . . . . . . . . . 14 |
18 | 14, 17 | mpdan 417 | . . . . . . . . . . . . 13 |
19 | 18 | impcom 124 | . . . . . . . . . . . 12 |
20 | 19 | reximi 2527 | . . . . . . . . . . 11 |
21 | 10, 20 | syl 14 | . . . . . . . . . 10 |
22 | 21 | ex 114 | . . . . . . . . 9 |
23 | 9, 22 | syl 14 | . . . . . . . 8 |
24 | 23 | adantll 467 | . . . . . . 7 |
25 | 3, 24 | orim12d 775 | . . . . . 6 |
26 | 25 | ex 114 | . . . . 5 |
27 | 1, 26 | syl7bi 164 | . . . 4 |
28 | 27 | alrimiv 1846 | . . 3 |
29 | nfv 1508 | . . . . 5 | |
30 | bj-findis.nf1 | . . . . 5 | |
31 | 29, 30 | nfim 1551 | . . . 4 |
32 | nfv 1508 | . . . . 5 | |
33 | nfv 1508 | . . . . . . 7 | |
34 | bj-findis.nf0 | . . . . . . 7 | |
35 | 33, 34 | nfan 1544 | . . . . . 6 |
36 | nfcv 2279 | . . . . . . 7 | |
37 | nfv 1508 | . . . . . . . 8 | |
38 | bj-findis.nfsuc | . . . . . . . 8 | |
39 | 37, 38 | nfan 1544 | . . . . . . 7 |
40 | 36, 39 | nfrexxy 2470 | . . . . . 6 |
41 | 35, 40 | nfor 1553 | . . . . 5 |
42 | 32, 41 | nfim 1551 | . . . 4 |
43 | nfv 1508 | . . . 4 | |
44 | nfv 1508 | . . . 4 | |
45 | eleq1 2200 | . . . . . 6 | |
46 | 45 | biimprd 157 | . . . . 5 |
47 | bj-findis.1 | . . . . 5 | |
48 | 46, 47 | imim12d 74 | . . . 4 |
49 | eleq1 2200 | . . . . . 6 | |
50 | 49 | biimpd 143 | . . . . 5 |
51 | eqtr 2155 | . . . . . . . 8 | |
52 | bj-findis.0 | . . . . . . . 8 | |
53 | 51, 52 | syl 14 | . . . . . . 7 |
54 | 53 | expimpd 360 | . . . . . 6 |
55 | eqtr 2155 | . . . . . . . . 9 | |
56 | bj-findis.suc | . . . . . . . . 9 | |
57 | 55, 56 | syl 14 | . . . . . . . 8 |
58 | 57 | expimpd 360 | . . . . . . 7 |
59 | 58 | rexlimdvw 2551 | . . . . . 6 |
60 | 54, 59 | jaod 706 | . . . . 5 |
61 | 50, 60 | imim12d 74 | . . . 4 |
62 | 31, 42, 43, 44, 48, 61 | setindis 13154 | . . 3 |
63 | 28, 62 | syl 14 | . 2 |
64 | df-ral 2419 | . 2 | |
65 | 63, 64 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wal 1329 wceq 1331 wnf 1436 wcel 1480 wral 2414 wrex 2415 c0 3358 csuc 4282 com 4499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-nul 4049 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-bd0 13000 ax-bdim 13001 ax-bdan 13002 ax-bdor 13003 ax-bdn 13004 ax-bdal 13005 ax-bdex 13006 ax-bdeq 13007 ax-bdel 13008 ax-bdsb 13009 ax-bdsep 13071 ax-infvn 13128 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-sn 3528 df-pr 3529 df-uni 3732 df-int 3767 df-suc 4288 df-iom 4500 df-bdc 13028 df-bj-ind 13114 |
This theorem is referenced by: bj-findisg 13167 bj-findes 13168 |
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