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Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 13166 for explanations. From this version, it is easy to prove findes 4512. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-findes |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . . . 4 | |
2 | nfv 1508 | . . . 4 | |
3 | 1, 2 | nfim 1551 | . . 3 |
4 | nfs1v 1910 | . . . 4 | |
5 | nfsbc1v 2922 | . . . 4 | |
6 | 4, 5 | nfim 1551 | . . 3 |
7 | sbequ12 1744 | . . . 4 | |
8 | suceq 4319 | . . . . 5 | |
9 | 8 | sbceq1d 2909 | . . . 4 |
10 | 7, 9 | imbi12d 233 | . . 3 |
11 | 3, 6, 10 | cbvral 2648 | . 2 |
12 | nfsbc1v 2922 | . . 3 | |
13 | sbceq1a 2913 | . . . 4 | |
14 | 13 | biimprd 157 | . . 3 |
15 | sbequ1 1741 | . . 3 | |
16 | sbceq1a 2913 | . . . 4 | |
17 | 16 | biimprd 157 | . . 3 |
18 | 12, 4, 5, 14, 15, 17 | bj-findis 13166 | . 2 |
19 | 11, 18 | sylan2b 285 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wsb 1735 wral 2414 wsbc 2904 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-sbc 2905 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: (None) |
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