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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-bdfindes | GIF version |
Description: Bounded induction (principle of induction for bounded formulas), using explicit substitutions. Constructive proof (from CZF). See the comment of bj-bdfindis 13593 for explanations. From this version, it is easy to prove the bounded version of findes 4564. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-bdfindes.bd | ⊢ BOUNDED 𝜑 |
Ref | Expression |
---|---|
bj-bdfindes | ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1508 | . . . 4 ⊢ Ⅎ𝑦[suc 𝑥 / 𝑥]𝜑 | |
3 | 1, 2 | nfim 1552 | . . 3 ⊢ Ⅎ𝑦(𝜑 → [suc 𝑥 / 𝑥]𝜑) |
4 | nfs1v 1919 | . . . 4 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
5 | nfsbc1v 2955 | . . . 4 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑 | |
6 | 4, 5 | nfim 1552 | . . 3 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑) |
7 | sbequ12 1751 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
8 | suceq 4364 | . . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦) | |
9 | 8 | sbceq1d 2942 | . . . 4 ⊢ (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) |
10 | 7, 9 | imbi12d 233 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))) |
11 | 3, 6, 10 | cbvral 2676 | . 2 ⊢ (∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) |
12 | bj-bdfindes.bd | . . 3 ⊢ BOUNDED 𝜑 | |
13 | nfsbc1v 2955 | . . 3 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑 | |
14 | sbceq1a 2946 | . . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑)) | |
15 | 14 | biimprd 157 | . . 3 ⊢ (𝑥 = ∅ → ([∅ / 𝑥]𝜑 → 𝜑)) |
16 | sbequ1 1748 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
17 | sbceq1a 2946 | . . . 4 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) | |
18 | 17 | biimprd 157 | . . 3 ⊢ (𝑥 = suc 𝑦 → ([suc 𝑦 / 𝑥]𝜑 → 𝜑)) |
19 | 12, 13, 4, 5, 15, 16, 18 | bj-bdfindis 13593 | . 2 ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) |
20 | 11, 19 | sylan2b 285 | 1 ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1335 [wsb 1742 ∀wral 2435 [wsbc 2937 ∅c0 3395 suc csuc 4327 ωcom 4551 BOUNDED wbd 13458 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-nul 4092 ax-pr 4171 ax-un 4395 ax-bd0 13459 ax-bdor 13462 ax-bdex 13465 ax-bdeq 13466 ax-bdel 13467 ax-bdsb 13468 ax-bdsep 13530 ax-infvn 13587 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-sn 3567 df-pr 3568 df-uni 3775 df-int 3810 df-suc 4333 df-iom 4552 df-bdc 13487 df-bj-ind 13573 |
This theorem is referenced by: (None) |
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