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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 13702 for explanations. From this version, it is easy to prove findes 4574. (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 1515 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1515 | . . . 4 ⊢ Ⅎ𝑦[suc 𝑥 / 𝑥]𝜑 | |
3 | 1, 2 | nfim 1559 | . . 3 ⊢ Ⅎ𝑦(𝜑 → [suc 𝑥 / 𝑥]𝜑) |
4 | nfs1v 1926 | . . . 4 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
5 | nfsbc1v 2964 | . . . 4 ⊢ Ⅎ𝑥[suc 𝑦 / 𝑥]𝜑 | |
6 | 4, 5 | nfim 1559 | . . 3 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑) |
7 | sbequ12 1758 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
8 | suceq 4374 | . . . . 5 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦) | |
9 | 8 | sbceq1d 2951 | . . . 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 2685 | . 2 ⊢ (∀𝑥 ∈ ω (𝜑 → [suc 𝑥 / 𝑥]𝜑) ↔ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) |
12 | nfsbc1v 2964 | . . 3 ⊢ Ⅎ𝑥[∅ / 𝑥]𝜑 | |
13 | sbceq1a 2955 | . . . 4 ⊢ (𝑥 = ∅ → (𝜑 ↔ [∅ / 𝑥]𝜑)) | |
14 | 13 | biimprd 157 | . . 3 ⊢ (𝑥 = ∅ → ([∅ / 𝑥]𝜑 → 𝜑)) |
15 | sbequ1 1755 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → [𝑦 / 𝑥]𝜑)) | |
16 | sbceq1a 2955 | . . . 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 13702 | . 2 ⊢ (([∅ / 𝑥]𝜑 ∧ ∀𝑦 ∈ ω ([𝑦 / 𝑥]𝜑 → [suc 𝑦 / 𝑥]𝜑)) → ∀𝑥 ∈ ω 𝜑) |
19 | 11, 18 | 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 1342 [wsb 1749 ∀wral 2442 [wsbc 2946 ∅c0 3404 suc csuc 4337 ωcom 4561 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-nul 4102 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-bd0 13536 ax-bdim 13537 ax-bdan 13538 ax-bdor 13539 ax-bdn 13540 ax-bdal 13541 ax-bdex 13542 ax-bdeq 13543 ax-bdel 13544 ax-bdsb 13545 ax-bdsep 13607 ax-infvn 13664 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-suc 4343 df-iom 4562 df-bdc 13564 df-bj-ind 13650 |
This theorem is referenced by: (None) |
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