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Mirrors > Home > MPE Home > Th. List > nfrald | Structured version Visualization version GIF version |
Description: Deduction version of nfral 3228. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker nfraldw 3225 when possible. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfrald.1 | ⊢ Ⅎ𝑦𝜑 |
nfrald.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfrald.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfrald | ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3145 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | |
2 | nfrald.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvf 3009 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
4 | 3 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦) |
5 | nfrald.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | 5 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴) |
7 | 4, 6 | nfeld 2991 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴) |
8 | nfrald.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
9 | 8 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
10 | 7, 9 | nfimd 1895 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)) |
11 | 2, 10 | nfald2 2467 | . 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) |
12 | 1, 11 | nfxfrd 1854 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1535 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2963 ∀wral 3140 |
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-13 2390 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 |
This theorem is referenced by: nfral 3228 nfrexdg 3310 |
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