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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-findes | Unicode version |
Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 13348 for explanations. From this version, it is easy to prove findes 4525. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-findes |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1509 |
. . . 4
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2 | nfv 1509 |
. . . 4
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3 | 1, 2 | nfim 1552 |
. . 3
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4 | nfs1v 1913 |
. . . 4
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5 | nfsbc1v 2931 |
. . . 4
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6 | 4, 5 | nfim 1552 |
. . 3
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7 | sbequ12 1745 |
. . . 4
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8 | suceq 4332 |
. . . . 5
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9 | 8 | sbceq1d 2918 |
. . . 4
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10 | 7, 9 | imbi12d 233 |
. . 3
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11 | 3, 6, 10 | cbvral 2653 |
. 2
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12 | nfsbc1v 2931 |
. . 3
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13 | sbceq1a 2922 |
. . . 4
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14 | 13 | biimprd 157 |
. . 3
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15 | sbequ1 1742 |
. . 3
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16 | sbceq1a 2922 |
. . . 4
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17 | 16 | biimprd 157 |
. . 3
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18 | 12, 4, 5, 14, 15, 17 | bj-findis 13348 |
. 2
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19 | 11, 18 | sylan2b 285 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-nul 4062 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-bd0 13182 ax-bdim 13183 ax-bdan 13184 ax-bdor 13185 ax-bdn 13186 ax-bdal 13187 ax-bdex 13188 ax-bdeq 13189 ax-bdel 13190 ax-bdsb 13191 ax-bdsep 13253 ax-infvn 13310 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-suc 4301 df-iom 4513 df-bdc 13210 df-bj-ind 13296 |
This theorem is referenced by: (None) |
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