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Mirrors > Home > ILE Home > Th. List > Mathboxes > setindis | GIF version |
Description: Axiom of set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) |
Ref | Expression |
---|---|
setindis.nf0 | ⊢ Ⅎ𝑥𝜓 |
setindis.nf1 | ⊢ Ⅎ𝑥𝜒 |
setindis.nf2 | ⊢ Ⅎ𝑦𝜑 |
setindis.nf3 | ⊢ Ⅎ𝑦𝜓 |
setindis.1 | ⊢ (𝑥 = 𝑧 → (𝜑 → 𝜓)) |
setindis.2 | ⊢ (𝑥 = 𝑦 → (𝜒 → 𝜑)) |
Ref | Expression |
---|---|
setindis | ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2282 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
2 | setindis.nf0 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
3 | 1, 2 | nfralxy 2474 | . . . 4 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 𝜓 |
4 | setindis.nf1 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
5 | 3, 4 | nfim 1552 | . . 3 ⊢ Ⅎ𝑥(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) |
6 | nfcv 2282 | . . . . 5 ⊢ Ⅎ𝑦𝑥 | |
7 | setindis.nf3 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
8 | 6, 7 | nfralxy 2474 | . . . 4 ⊢ Ⅎ𝑦∀𝑧 ∈ 𝑥 𝜓 |
9 | setindis.nf2 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
10 | 8, 9 | nfim 1552 | . . 3 ⊢ Ⅎ𝑦(∀𝑧 ∈ 𝑥 𝜓 → 𝜑) |
11 | raleq 2629 | . . . . 5 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑦 𝜓 ↔ ∀𝑧 ∈ 𝑥 𝜓)) | |
12 | 11 | biimprd 157 | . . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑥 𝜓 → ∀𝑧 ∈ 𝑦 𝜓)) |
13 | setindis.2 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜒 → 𝜑)) | |
14 | 13 | equcoms 1685 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝜒 → 𝜑)) |
15 | 12, 14 | imim12d 74 | . . 3 ⊢ (𝑦 = 𝑥 → ((∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → (∀𝑧 ∈ 𝑥 𝜓 → 𝜑))) |
16 | 5, 10, 15 | cbv3 1721 | . 2 ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥(∀𝑧 ∈ 𝑥 𝜓 → 𝜑)) |
17 | setindis.1 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝜑 → 𝜓)) | |
18 | 2, 17 | bj-sbime 13151 | . . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 → 𝜓) |
19 | 18 | ralimi 2498 | . . . 4 ⊢ (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 → ∀𝑧 ∈ 𝑥 𝜓) |
20 | 19 | imim1i 60 | . . 3 ⊢ ((∀𝑧 ∈ 𝑥 𝜓 → 𝜑) → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 → 𝜑)) |
21 | 20 | alimi 1432 | . 2 ⊢ (∀𝑥(∀𝑧 ∈ 𝑥 𝜓 → 𝜑) → ∀𝑥(∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 → 𝜑)) |
22 | ax-setind 4460 | . 2 ⊢ (∀𝑥(∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 → 𝜑) → ∀𝑥𝜑) | |
23 | 16, 21, 22 | 3syl 17 | 1 ⊢ (∀𝑦(∀𝑧 ∈ 𝑦 𝜓 → 𝜒) → ∀𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1330 Ⅎwnf 1437 [wsb 1736 ∀wral 2417 |
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-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-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 |
This theorem is referenced by: bj-inf2vnlem4 13342 bj-findis 13348 |
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