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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdsetindis | GIF version |
Description: Axiom of bounded set induction using implicit substitutions. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bdsetindis.bd | ⊢ BOUNDED φ |
bdsetindis.nf0 | ⊢ Ⅎxψ |
bdsetindis.nf1 | ⊢ Ⅎxχ |
bdsetindis.nf2 | ⊢ Ⅎyφ |
bdsetindis.nf3 | ⊢ Ⅎyψ |
bdsetindis.1 | ⊢ (x = z → (φ → ψ)) |
bdsetindis.2 | ⊢ (x = y → (χ → φ)) |
Ref | Expression |
---|---|
bdsetindis | ⊢ (∀y(∀z ∈ y ψ → χ) → ∀xφ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2175 | . . . . 5 ⊢ Ⅎxy | |
2 | bdsetindis.nf0 | . . . . 5 ⊢ Ⅎxψ | |
3 | 1, 2 | nfralxy 2354 | . . . 4 ⊢ Ⅎx∀z ∈ y ψ |
4 | bdsetindis.nf1 | . . . 4 ⊢ Ⅎxχ | |
5 | 3, 4 | nfim 1461 | . . 3 ⊢ Ⅎx(∀z ∈ y ψ → χ) |
6 | nfcv 2175 | . . . . 5 ⊢ Ⅎyx | |
7 | bdsetindis.nf3 | . . . . 5 ⊢ Ⅎyψ | |
8 | 6, 7 | nfralxy 2354 | . . . 4 ⊢ Ⅎy∀z ∈ x ψ |
9 | bdsetindis.nf2 | . . . 4 ⊢ Ⅎyφ | |
10 | 8, 9 | nfim 1461 | . . 3 ⊢ Ⅎy(∀z ∈ x ψ → φ) |
11 | raleq 2499 | . . . . 5 ⊢ (y = x → (∀z ∈ y ψ ↔ ∀z ∈ x ψ)) | |
12 | 11 | biimprd 147 | . . . 4 ⊢ (y = x → (∀z ∈ x ψ → ∀z ∈ y ψ)) |
13 | bdsetindis.2 | . . . . 5 ⊢ (x = y → (χ → φ)) | |
14 | 13 | equcoms 1591 | . . . 4 ⊢ (y = x → (χ → φ)) |
15 | 12, 14 | imim12d 68 | . . 3 ⊢ (y = x → ((∀z ∈ y ψ → χ) → (∀z ∈ x ψ → φ))) |
16 | 5, 10, 15 | cbv3 1627 | . 2 ⊢ (∀y(∀z ∈ y ψ → χ) → ∀x(∀z ∈ x ψ → φ)) |
17 | bdsetindis.1 | . . . . . 6 ⊢ (x = z → (φ → ψ)) | |
18 | 2, 17 | bj-sbime 9248 | . . . . 5 ⊢ ([z / x]φ → ψ) |
19 | 18 | ralimi 2378 | . . . 4 ⊢ (∀z ∈ x [z / x]φ → ∀z ∈ x ψ) |
20 | 19 | imim1i 54 | . . 3 ⊢ ((∀z ∈ x ψ → φ) → (∀z ∈ x [z / x]φ → φ)) |
21 | 20 | alimi 1341 | . 2 ⊢ (∀x(∀z ∈ x ψ → φ) → ∀x(∀z ∈ x [z / x]φ → φ)) |
22 | bdsetindis.bd | . . 3 ⊢ BOUNDED φ | |
23 | 22 | ax-bdsetind 9428 | . 2 ⊢ (∀x(∀z ∈ x [z / x]φ → φ) → ∀xφ) |
24 | 16, 21, 23 | 3syl 17 | 1 ⊢ (∀y(∀z ∈ y ψ → χ) → ∀xφ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1240 Ⅎwnf 1346 [wsb 1642 ∀wral 2300 BOUNDED wbd 9267 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-bdsetind 9428 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 |
This theorem is referenced by: bj-inf2vnlem3 9432 |
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