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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-bdfindes | Unicode version |
Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13072 for explanations. From this version, it is easy to prove the bounded version of findes 4487. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-bdfindes.bd | BOUNDED |
Ref | Expression |
---|---|
bj-bdfindes |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1493 | . . . 4 | |
2 | nfv 1493 | . . . 4 | |
3 | 1, 2 | nfim 1536 | . . 3 |
4 | nfs1v 1892 | . . . 4 | |
5 | nfsbc1v 2900 | . . . 4 | |
6 | 4, 5 | nfim 1536 | . . 3 |
7 | sbequ12 1729 | . . . 4 | |
8 | suceq 4294 | . . . . 5 | |
9 | 8 | sbceq1d 2887 | . . . 4 |
10 | 7, 9 | imbi12d 233 | . . 3 |
11 | 3, 6, 10 | cbvral 2627 | . 2 |
12 | bj-bdfindes.bd | . . 3 BOUNDED | |
13 | nfsbc1v 2900 | . . 3 | |
14 | sbceq1a 2891 | . . . 4 | |
15 | 14 | biimprd 157 | . . 3 |
16 | sbequ1 1726 | . . 3 | |
17 | sbceq1a 2891 | . . . 4 | |
18 | 17 | biimprd 157 | . . 3 |
19 | 12, 13, 4, 5, 15, 16, 18 | bj-bdfindis 13072 | . 2 |
20 | 11, 19 | sylan2b 285 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wsb 1720 wral 2393 wsbc 2882 c0 3333 csuc 4257 com 4474 BOUNDED wbd 12937 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-nul 4024 ax-pr 4101 ax-un 4325 ax-bd0 12938 ax-bdor 12941 ax-bdex 12944 ax-bdeq 12945 ax-bdel 12946 ax-bdsb 12947 ax-bdsep 13009 ax-infvn 13066 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-suc 4263 df-iom 4475 df-bdc 12966 df-bj-ind 13052 |
This theorem is referenced by: (None) |
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