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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 15209 for explanations. From this version, it is easy to prove findes 4620. (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 1539 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1539 | . . . 4 ⊢ Ⅎ𝑦[suc 𝑥 / 𝑥]𝜑 | |
3 | 1, 2 | nfim 1583 | . . 3 ⊢ Ⅎ𝑦(𝜑 → [suc 𝑥 / 𝑥]𝜑) |
4 | nfs1v 1951 | . . . 4 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
5 | nfsbc1v 2996 | . . . 4 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑 | |
6 | 4, 5 | nfim 1583 | . . 3 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑) |
7 | sbequ12 1782 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
8 | suceq 4420 | . . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦) | |
9 | 8 | sbceq1d 2982 | . . . 4 ⊢ (𝑥 = 𝑦 → ([suc 𝑥 / 𝑥]𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) |
10 | 7, 9 | imbi12d 234 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑))) |
11 | 3, 6, 10 | cbvral 2714 | . 2 ⊢ (∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) |
12 | nfsbc1v 2996 | . . 3 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑 | |
13 | sbceq1a 2987 | . . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑)) | |
14 | 13 | biimprd 158 | . . 3 ⊢ (𝑥 = ∅ → ([∅ / 𝑥]𝜑 → 𝜑)) |
15 | sbequ1 1779 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
16 | sbceq1a 2987 | . . . 4 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ [suc 𝑦 / 𝑥]𝜑)) | |
17 | 16 | biimprd 158 | . . 3 ⊢ (𝑥 = suc 𝑦 → ([suc 𝑦 / 𝑥]𝜑 → 𝜑)) |
18 | 12, 4, 5, 14, 15, 17 | bj-findis 15209 | . 2 ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) |
19 | 11, 18 | sylan2b 287 | 1 ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 [wsb 1773 ∀wral 2468 [wsbc 2977 ∅c0 3437 suc csuc 4383 ωcom 4607 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-nul 4144 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-bd0 15043 ax-bdim 15044 ax-bdan 15045 ax-bdor 15046 ax-bdn 15047 ax-bdal 15048 ax-bdex 15049 ax-bdeq 15050 ax-bdel 15051 ax-bdsb 15052 ax-bdsep 15114 ax-infvn 15171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-suc 4389 df-iom 4608 df-bdc 15071 df-bj-ind 15157 |
This theorem is referenced by: (None) |
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