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Mirrors > Home > MPE Home > Th. List > dvelimf | Structured version Visualization version GIF version |
Description: Version of dvelimv 2451 without any variable restrictions. Usage of this theorem is discouraged because it depends on ax-13 2371. (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 2416 | . . 3 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ 𝜓) |
4 | 3 | bicomi 223 | . 2 ⊢ (𝜓 ↔ ∀𝑧(𝑧 = 𝑦 → 𝜑)) |
5 | nfnae 2433 | . . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑥 𝑥 = 𝑦 | |
6 | nfeqf 2380 | . . . . 5 ⊢ ((¬ ∀𝑥 𝑥 = 𝑧 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑧 = 𝑦) | |
7 | 6 | ancoms 460 | . . . 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 1898 | . . 3 ⊢ ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑧 = 𝑦 → 𝜑)) |
11 | 5, 10 | nfald2 2444 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥∀𝑧(𝑧 = 𝑦 → 𝜑)) |
12 | 4, 11 | nfxfrd 1857 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 Ⅎwnf 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 |
This theorem is referenced by: dvelimdf 2448 dvelimh 2449 dvelimnf 2452 |
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