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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 13982 for explanations. From this version, it is easy to prove the bounded version of findes 4587. (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 1521 | . . . 4 | |
2 | nfv 1521 | . . . 4 | |
3 | 1, 2 | nfim 1565 | . . 3 |
4 | nfs1v 1932 | . . . 4 | |
5 | nfsbc1v 2973 | . . . 4 | |
6 | 4, 5 | nfim 1565 | . . 3 |
7 | sbequ12 1764 | . . . 4 | |
8 | suceq 4387 | . . . . 5 | |
9 | 8 | sbceq1d 2960 | . . . 4 |
10 | 7, 9 | imbi12d 233 | . . 3 |
11 | 3, 6, 10 | cbvral 2692 | . 2 |
12 | bj-bdfindes.bd | . . 3 BOUNDED | |
13 | nfsbc1v 2973 | . . 3 | |
14 | sbceq1a 2964 | . . . 4 | |
15 | 14 | biimprd 157 | . . 3 |
16 | sbequ1 1761 | . . 3 | |
17 | sbceq1a 2964 | . . . 4 | |
18 | 17 | biimprd 157 | . . 3 |
19 | 12, 13, 4, 5, 15, 16, 18 | bj-bdfindis 13982 | . 2 |
20 | 11, 19 | sylan2b 285 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wsb 1755 wral 2448 wsbc 2955 c0 3414 csuc 4350 com 4574 BOUNDED wbd 13847 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-nul 4115 ax-pr 4194 ax-un 4418 ax-bd0 13848 ax-bdor 13851 ax-bdex 13854 ax-bdeq 13855 ax-bdel 13856 ax-bdsb 13857 ax-bdsep 13919 ax-infvn 13976 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-sn 3589 df-pr 3590 df-uni 3797 df-int 3832 df-suc 4356 df-iom 4575 df-bdc 13876 df-bj-ind 13962 |
This theorem is referenced by: (None) |
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