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Mirrors > Home > MPE Home > Th. List > nfim1 | Structured version Visualization version GIF version |
Description: A closed form of nfim 1813. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
Ref | Expression |
---|---|
nfim1.1 | ⊢ Ⅎ𝑥𝜑 |
nfim1.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfim1 | ⊢ Ⅎ𝑥(𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfim1.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | nf3 1703 | . . 3 ⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑)) | |
3 | 1, 2 | mpbi 219 | . 2 ⊢ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) |
4 | nftht0 1709 | . . . 4 ⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑) | |
5 | nfim1.2 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
6 | 5 | sps 2043 | . . . 4 ⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜓) |
7 | 4, 6 | nfimd 1812 | . . 3 ⊢ (∀𝑥𝜑 → Ⅎ𝑥(𝜑 → 𝜓)) |
8 | pm2.21 119 | . . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓)) | |
9 | 8 | alimi 1730 | . . . 4 ⊢ (∀𝑥 ¬ 𝜑 → ∀𝑥(𝜑 → 𝜓)) |
10 | nftht0 1709 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → Ⅎ𝑥(𝜑 → 𝜓)) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥(𝜑 → 𝜓)) |
12 | 7, 11 | jaoi 393 | . 2 ⊢ ((∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑) → Ⅎ𝑥(𝜑 → 𝜓)) |
13 | 3, 12 | ax-mp 5 | 1 ⊢ Ⅎ𝑥(𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∀wal 1473 Ⅎwnf 1699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-or 384 df-ex 1696 df-nf 1701 |
This theorem is referenced by: nfan1 2056 cbv1 2255 dvelimdf 2323 sbied 2397 sbco2d 2404 bj-cbv1v 31916 |
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