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Mirrors > Home > MPE Home > Th. List > dvelimf | Structured version Visualization version GIF version |
Description: Version of dvelimv 2473 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dvelimf.1 | ⊢ Ⅎ𝑥𝜑 |
dvelimf.2 | ⊢ Ⅎ𝑧𝜓 |
dvelimf.3 | ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
dvelimf | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvelimf.2 | . . . 4 ⊢ Ⅎ𝑧𝜓 | |
2 | dvelimf.3 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | equsal 2438 | . . 3 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ 𝜓) |
4 | 3 | bicomi 226 | . 2 ⊢ (𝜓 ↔ ∀𝑧(𝑧 = 𝑦 → 𝜑)) |
5 | nfnae 2455 | . . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦 | |
6 | nfeqf 2398 | . . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦) | |
7 | 6 | ancoms 461 | . . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑧 = 𝑦) |
8 | dvelimf.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
9 | 8 | a1i 11 | . . . 4 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝜑) |
10 | 7, 9 | nfimd 1894 | . . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑧 = 𝑦 → 𝜑)) |
11 | 5, 10 | nfald2 2466 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜑)) |
12 | 4, 11 | nfxfrd 1853 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 Ⅎwnf 1783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-10 2144 ax-11 2160 ax-12 2176 ax-13 2389 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 |
This theorem is referenced by: dvelimdf 2470 dvelimh 2471 dvelimnf 2474 |
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