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Mirrors > Home > ILE Home > Th. List > Mathboxes > setindft | GIF version |
Description: Axiom of set-induction with a DV condition replaced with a non-freeness hypothesis (Contributed by BJ, 22-Nov-2019.) |
Ref | Expression |
---|---|
setindft | ⊢ (∀xℲyφ → (∀x(∀y ∈ x [y / x]φ → φ) → ∀xφ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 1431 | . . 3 ⊢ Ⅎx∀xℲyφ | |
2 | nfv 1418 | . . . . . 6 ⊢ Ⅎz∀xℲyφ | |
3 | nfnf1 1433 | . . . . . . 7 ⊢ ℲyℲyφ | |
4 | 3 | nfal 1465 | . . . . . 6 ⊢ Ⅎy∀xℲyφ |
5 | nfsbt 1847 | . . . . . 6 ⊢ (∀xℲyφ → Ⅎy[z / x]φ) | |
6 | nfv 1418 | . . . . . . 7 ⊢ Ⅎz[y / x]φ | |
7 | 6 | a1i 9 | . . . . . 6 ⊢ (∀xℲyφ → Ⅎz[y / x]φ) |
8 | sbequ 1718 | . . . . . . 7 ⊢ (z = y → ([z / x]φ ↔ [y / x]φ)) | |
9 | 8 | a1i 9 | . . . . . 6 ⊢ (∀xℲyφ → (z = y → ([z / x]φ ↔ [y / x]φ))) |
10 | 2, 4, 5, 7, 9 | cbvrald 9262 | . . . . 5 ⊢ (∀xℲyφ → (∀z ∈ x [z / x]φ ↔ ∀y ∈ x [y / x]φ)) |
11 | 10 | biimpd 132 | . . . 4 ⊢ (∀xℲyφ → (∀z ∈ x [z / x]φ → ∀y ∈ x [y / x]φ)) |
12 | 11 | imim1d 69 | . . 3 ⊢ (∀xℲyφ → ((∀y ∈ x [y / x]φ → φ) → (∀z ∈ x [z / x]φ → φ))) |
13 | 1, 12 | alimd 1411 | . 2 ⊢ (∀xℲyφ → (∀x(∀y ∈ x [y / x]φ → φ) → ∀x(∀z ∈ x [z / x]φ → φ))) |
14 | ax-setind 4220 | . 2 ⊢ (∀x(∀z ∈ x [z / x]φ → φ) → ∀xφ) | |
15 | 13, 14 | syl6 29 | 1 ⊢ (∀xℲyφ → (∀x(∀y ∈ x [y / x]φ → φ) → ∀xφ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1240 Ⅎwnf 1346 [wsb 1642 ∀wral 2300 |
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-setind 4220 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-cleq 2030 df-clel 2033 df-ral 2305 |
This theorem is referenced by: setindf 9426 |
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