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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 10072 for a bounded version not requiring ax-setind 4262. See finds 4323 for a proof in IZF. From this version, it is easy to prove of finds 4323, finds2 4324, finds1 4325. (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 10089 | . . . . 5 | |
2 | pm3.21 251 | . . . . . . . 8 | |
3 | 2 | ad2antrr 457 | . . . . . . 7 |
4 | pm2.04 76 | . . . . . . . . . . 11 | |
5 | 4 | ralimi2 2381 | . . . . . . . . . 10 |
6 | imim2 49 | . . . . . . . . . . . 12 | |
7 | 6 | ral2imi 2385 | . . . . . . . . . . 11 |
8 | 7 | imp 115 | . . . . . . . . . 10 |
9 | 5, 8 | sylan2 270 | . . . . . . . . 9 |
10 | r19.29 2450 | . . . . . . . . . . 11 | |
11 | vex 2560 | . . . . . . . . . . . . . . . 16 | |
12 | 11 | sucid 4154 | . . . . . . . . . . . . . . 15 |
13 | eleq2 2101 | . . . . . . . . . . . . . . 15 | |
14 | 12, 13 | mpbiri 157 | . . . . . . . . . . . . . 14 |
15 | ax-1 5 | . . . . . . . . . . . . . . 15 | |
16 | pm2.27 35 | . . . . . . . . . . . . . . 15 | |
17 | 15, 16 | anim12ii 325 | . . . . . . . . . . . . . 14 |
18 | 14, 17 | mpdan 398 | . . . . . . . . . . . . 13 |
19 | 18 | impcom 116 | . . . . . . . . . . . 12 |
20 | 19 | reximi 2416 | . . . . . . . . . . 11 |
21 | 10, 20 | syl 14 | . . . . . . . . . 10 |
22 | 21 | ex 108 | . . . . . . . . 9 |
23 | 9, 22 | syl 14 | . . . . . . . 8 |
24 | 23 | adantll 445 | . . . . . . 7 |
25 | 3, 24 | orim12d 700 | . . . . . 6 |
26 | 25 | ex 108 | . . . . 5 |
27 | 1, 26 | syl7bi 154 | . . . 4 |
28 | 27 | alrimiv 1754 | . . 3 |
29 | nfv 1421 | . . . . 5 | |
30 | bj-findis.nf1 | . . . . 5 | |
31 | 29, 30 | nfim 1464 | . . . 4 |
32 | nfv 1421 | . . . . 5 | |
33 | nfv 1421 | . . . . . . 7 | |
34 | bj-findis.nf0 | . . . . . . 7 | |
35 | 33, 34 | nfan 1457 | . . . . . 6 |
36 | nfcv 2178 | . . . . . . 7 | |
37 | nfv 1421 | . . . . . . . 8 | |
38 | bj-findis.nfsuc | . . . . . . . 8 | |
39 | 37, 38 | nfan 1457 | . . . . . . 7 |
40 | 36, 39 | nfrexxy 2361 | . . . . . 6 |
41 | 35, 40 | nfor 1466 | . . . . 5 |
42 | 32, 41 | nfim 1464 | . . . 4 |
43 | nfv 1421 | . . . 4 | |
44 | nfv 1421 | . . . 4 | |
45 | eleq1 2100 | . . . . . 6 | |
46 | 45 | biimprd 147 | . . . . 5 |
47 | bj-findis.1 | . . . . 5 | |
48 | 46, 47 | imim12d 68 | . . . 4 |
49 | eleq1 2100 | . . . . . 6 | |
50 | 49 | biimpd 132 | . . . . 5 |
51 | eqtr 2057 | . . . . . . . 8 | |
52 | bj-findis.0 | . . . . . . . 8 | |
53 | 51, 52 | syl 14 | . . . . . . 7 |
54 | 53 | expimpd 345 | . . . . . 6 |
55 | eqtr 2057 | . . . . . . . . 9 | |
56 | bj-findis.suc | . . . . . . . . 9 | |
57 | 55, 56 | syl 14 | . . . . . . . 8 |
58 | 57 | expimpd 345 | . . . . . . 7 |
59 | 58 | rexlimdvw 2436 | . . . . . 6 |
60 | 54, 59 | jaod 637 | . . . . 5 |
61 | 50, 60 | imim12d 68 | . . . 4 |
62 | 31, 42, 43, 44, 48, 61 | setindis 10092 | . . 3 |
63 | 28, 62 | syl 14 | . 2 |
64 | df-ral 2311 | . 2 | |
65 | 63, 64 | sylibr 137 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wo 629 wal 1241 wceq 1243 wnf 1349 wcel 1393 wral 2306 wrex 2307 c0 3224 csuc 4102 com 4313 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-nul 3883 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-bd0 9933 ax-bdim 9934 ax-bdan 9935 ax-bdor 9936 ax-bdn 9937 ax-bdal 9938 ax-bdex 9939 ax-bdeq 9940 ax-bdel 9941 ax-bdsb 9942 ax-bdsep 10004 ax-infvn 10066 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-iom 4314 df-bdc 9961 df-bj-ind 10051 |
This theorem is referenced by: bj-findisg 10105 bj-findes 10106 |
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