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Mirrors > Home > MPE Home > Th. List > nfraldw | Structured version Visualization version GIF version |
Description: Deduction version of nfralw 3225. Version of nfrald 3224 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 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 3143 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | |
2 | nfraldw.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvd 2978 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝑦) | |
4 | nfraldw.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 3, 4 | nfeld 2989 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
6 | nfraldw.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
7 | 5, 6 | nfimd 1895 | . . 3 ⊢ (𝜑 → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)) |
8 | 2, 7 | nfald 2347 | . 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) |
9 | 1, 8 | nfxfrd 1854 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 ∀wral 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 |
This theorem is referenced by: nfralw 3225 nfrexd 3307 |
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