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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-findes | GIF version |
Description: Principle of induction, using explicit substitutions. Constructive proof (from CZF). See the comment of bj-findis 12762 for explanations. From this version, it is easy to prove findes 4455. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-findes | ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1476 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1476 | . . . 4 ⊢ Ⅎ𝑦[suc 𝑥 / 𝑥]𝜑 | |
3 | 1, 2 | nfim 1519 | . . 3 ⊢ Ⅎ𝑦(𝜑 → [suc 𝑥 / 𝑥]𝜑) |
4 | nfs1v 1875 | . . . 4 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
5 | nfsbc1v 2880 | . . . 4 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑 | |
6 | 4, 5 | nfim 1519 | . . 3 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑) |
7 | sbequ12 1712 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
8 | suceq 4262 | . . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦) | |
9 | 8 | sbceq1d 2867 | . . . 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 2608 | . 2 ⊢ (∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) |
12 | nfsbc1v 2880 | . . 3 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑 | |
13 | sbceq1a 2871 | . . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑)) | |
14 | 13 | biimprd 157 | . . 3 ⊢ (𝑥 = ∅ → ([∅ / 𝑥]𝜑 → 𝜑)) |
15 | sbequ1 1709 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
16 | sbceq1a 2871 | . . . 4 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) | |
17 | 16 | biimprd 157 | . . 3 ⊢ (𝑥 = suc 𝑦 → ([suc 𝑦 / 𝑥]𝜑 → 𝜑)) |
18 | 12, 4, 5, 14, 15, 17 | bj-findis 12762 | . 2 ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) |
19 | 11, 18 | sylan2b 283 | 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 1299 [wsb 1703 ∀wral 2375 [wsbc 2862 ∅c0 3310 suc csuc 4225 ωcom 4442 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-nul 3994 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-bd0 12592 ax-bdim 12593 ax-bdan 12594 ax-bdor 12595 ax-bdn 12596 ax-bdal 12597 ax-bdex 12598 ax-bdeq 12599 ax-bdel 12600 ax-bdsb 12601 ax-bdsep 12663 ax-infvn 12724 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-sn 3480 df-pr 3481 df-uni 3684 df-int 3719 df-suc 4231 df-iom 4443 df-bdc 12620 df-bj-ind 12710 |
This theorem is referenced by: (None) |
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