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Mirrors > Home > ILE Home > Th. List > nfraldya | GIF version |
Description: Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfraldxy 2398 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.) |
Ref | Expression |
---|---|
nfraldya.2 | ⊢ Ⅎ𝑦𝜑 |
nfraldya.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfraldya.4 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfraldya | ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2353 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | |
2 | sbim 1868 | . . . . . 6 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 → 𝜓) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 → [𝑧 / 𝑦]𝜓)) | |
3 | clelsb3 2183 | . . . . . . 7 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴) | |
4 | 3 | imbi1i 236 | . . . . . 6 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐴 → [𝑧 / 𝑦]𝜓) ↔ (𝑧 ∈ 𝐴 → [𝑧 / 𝑦]𝜓)) |
5 | 2, 4 | bitri 182 | . . . . 5 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 → 𝜓) ↔ (𝑧 ∈ 𝐴 → [𝑧 / 𝑦]𝜓)) |
6 | 5 | albii 1399 | . . . 4 ⊢ (∀𝑧[𝑧 / 𝑦](𝑦 ∈ 𝐴 → 𝜓) ↔ ∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑦]𝜓)) |
7 | nfv 1461 | . . . . 5 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 → 𝜓) | |
8 | 7 | sb8 1777 | . . . 4 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝜓) ↔ ∀𝑧[𝑧 / 𝑦](𝑦 ∈ 𝐴 → 𝜓)) |
9 | df-ral 2353 | . . . 4 ⊢ (∀𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧(𝑧 ∈ 𝐴 → [𝑧 / 𝑦]𝜓)) | |
10 | 6, 8, 9 | 3bitr4i 210 | . . 3 ⊢ (∀𝑦(𝑦 ∈ 𝐴 → 𝜓) ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓) |
11 | nfv 1461 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
12 | nfraldya.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
13 | nfraldya.2 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
14 | nfraldya.4 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
15 | 13, 14 | nfsbd 1892 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓) |
16 | 11, 12, 15 | nfraldxy 2398 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓) |
17 | 10, 16 | nfxfrd 1404 | . 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) |
18 | 1, 17 | nfxfrd 1404 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 Ⅎwnf 1389 ∈ wcel 1433 [wsb 1685 Ⅎwnfc 2206 ∀wral 2348 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 |
This theorem is referenced by: nfralya 2404 |
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