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Mirrors > Home > ILE Home > Th. List > nfraldw | GIF version |
Description: Not-free for restricted universal quantification where 𝑥 and 𝑦 are distinct. See nfraldya 2529 for a version with 𝑦 and 𝐴 distinct instead. (Contributed by NM, 15-Feb-2013.) (Revised by GG, 10-Jan-2024.) |
Ref | Expression |
---|---|
nfraldw.1 | ⊢ Ⅎ𝑦𝜑 |
nfraldw.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfraldw.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfraldw | ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 2477 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | |
2 | nfraldw.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvd 2337 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) | |
4 | nfraldw.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 3, 4 | nfeld 2352 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
6 | nfraldw.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
7 | 5, 6 | nfimd 1596 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)) |
8 | 2, 7 | nfald 1771 | . 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) |
9 | 1, 8 | nfxfrd 1486 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1362 Ⅎwnf 1471 ∈ wcel 2164 Ⅎwnfc 2323 ∀wral 2472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-ial 1545 ax-i5r 1546 ax-ext 2175 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 |
This theorem is referenced by: nfralw 2531 |
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