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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-bdfindis | Unicode version |
Description: Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4596 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4596, finds2 4597, finds1 4598. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-bdfindis.bd |
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bj-bdfindis.nf0 |
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bj-bdfindis.nf1 |
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bj-bdfindis.nfsuc |
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bj-bdfindis.0 |
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bj-bdfindis.1 |
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bj-bdfindis.suc |
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Ref | Expression |
---|---|
bj-bdfindis |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-bdfindis.nf0 |
. . . 4
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2 | 0ex 4127 |
. . . 4
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3 | bj-bdfindis.0 |
. . . 4
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4 | 1, 2, 3 | elabf2 14190 |
. . 3
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5 | bj-bdfindis.nf1 |
. . . . . 6
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6 | bj-bdfindis.1 |
. . . . . 6
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7 | 5, 6 | elabf1 14189 |
. . . . 5
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8 | bj-bdfindis.nfsuc |
. . . . . 6
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9 | vex 2740 |
. . . . . . 7
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10 | 9 | bj-sucex 14331 |
. . . . . 6
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11 | bj-bdfindis.suc |
. . . . . 6
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12 | 8, 10, 11 | elabf2 14190 |
. . . . 5
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13 | 7, 12 | imim12i 59 |
. . . 4
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14 | 13 | ralimi 2540 |
. . 3
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15 | bj-bdfindis.bd |
. . . . 5
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16 | 15 | bdcab 14257 |
. . . 4
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17 | 16 | bdpeano5 14351 |
. . 3
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18 | 4, 14, 17 | syl2an 289 |
. 2
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19 | ssabral 3226 |
. 2
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20 | 18, 19 | sylib 122 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-nul 4126 ax-pr 4206 ax-un 4430 ax-bd0 14221 ax-bdor 14224 ax-bdex 14227 ax-bdeq 14228 ax-bdel 14229 ax-bdsb 14230 ax-bdsep 14292 ax-infvn 14349 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-sn 3597 df-pr 3598 df-uni 3808 df-int 3843 df-suc 4368 df-iom 4587 df-bdc 14249 df-bj-ind 14335 |
This theorem is referenced by: bj-bdfindisg 14356 bj-bdfindes 14357 bj-nn0suc0 14358 |
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