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Mirrors > Home > MPE Home > Th. List > nfraldw | Structured version Visualization version GIF version |
Description: Deduction version of nfralw 3308. Version of nfrald 3368 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 15-Feb-2013.) Avoid ax-9 2116, ax-ext 2703. (Revised by Gino Giotto, 24-Sep-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 3062 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | |
2 | nfraldw.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfraldw.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | 3 | nfcrd 2892 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
5 | nfraldw.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
6 | 4, 5 | nfimd 1897 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)) |
7 | 2, 6 | nfald 2321 | . 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) |
8 | 1, 7 | nfxfrd 1856 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-10 2137 ax-11 2154 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-nf 1786 df-clel 2810 df-nfc 2885 df-ral 3062 |
This theorem is referenced by: nfrexdw 3307 nfralwOLD 3309 nfttrcld 9704 |
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