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Mirrors > Home > MPE Home > Th. List > nfrald | Structured version Visualization version GIF version |
Description: Deduction version of nfral 3190. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker nfraldw 3187 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 3111 | . 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) | |
2 | nfrald.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvf 2981 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
4 | 3 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦) |
5 | nfrald.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | 5 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴) |
7 | 4, 6 | nfeld 2966 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴) |
8 | nfrald.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
9 | 8 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
10 | 7, 9 | nfimd 1895 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 → 𝜓)) |
11 | 2, 10 | nfald2 2456 | . 2 ⊢ (𝜑 → Ⅎ𝑥∀𝑦(𝑦 ∈ 𝐴 → 𝜓)) |
12 | 1, 11 | nfxfrd 1855 | 1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∀wal 1536 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-13 2379 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 |
This theorem is referenced by: nfral 3190 nfrexdg 3267 |
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