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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 1434 | . . 3 ⊢ Ⅎ𝑥∀𝑥Ⅎ𝑦𝜑 | |
2 | nfv 1421 | . . . . . 6 ⊢ Ⅎ𝑧∀𝑥Ⅎ𝑦𝜑 | |
3 | nfnf1 1436 | . . . . . . 7 ⊢ Ⅎ𝑦Ⅎ𝑦𝜑 | |
4 | 3 | nfal 1468 | . . . . . 6 ⊢ Ⅎ𝑦∀𝑥Ⅎ𝑦𝜑 |
5 | nfsbt 1850 | . . . . . 6 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑦[𝑧 / 𝑥]𝜑) | |
6 | nfv 1421 | . . . . . . 7 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | |
7 | 6 | a1i 9 | . . . . . 6 ⊢ (∀𝑥Ⅎ𝑦𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜑) |
8 | sbequ 1721 | . . . . . . 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 9927 | . . . . 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 1414 | . 2 ⊢ (∀𝑥Ⅎ𝑦𝜑 → (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑) → ∀𝑥(∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 → 𝜑))) |
14 | ax-setind 4262 | . 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 1241 Ⅎwnf 1349 [wsb 1645 ∀wral 2306 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-cleq 2033 df-clel 2036 df-ral 2311 |
This theorem is referenced by: setindf 10091 |
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