Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dvelimdf | GIF version |
Description: Deduction form of dvelimf 1932. This version may be useful if we want to avoid ax-17 1459 and use ax-16 1735 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) |
Ref | Expression |
---|---|
dvelimdf.1 | ⊢ Ⅎ𝑥𝜑 |
dvelimdf.2 | ⊢ Ⅎ𝑧𝜑 |
dvelimdf.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
dvelimdf.4 | ⊢ (𝜑 → Ⅎ𝑧𝜒) |
dvelimdf.5 | ⊢ (𝜑 → (𝑧 = 𝑦 → (𝜓 ↔ 𝜒))) |
Ref | Expression |
---|---|
dvelimdf | ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvelimdf.2 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
2 | dvelimdf.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
3 | 1, 2 | alrimi 1455 | . . 3 ⊢ (𝜑 → ∀𝑧Ⅎ𝑥𝜓) |
4 | nfsb4t 1931 | . . 3 ⊢ (∀𝑧Ⅎ𝑥𝜓 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑧]𝜓)) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑧]𝜓)) |
6 | dvelimdf.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
7 | dvelimdf.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑧𝜒) | |
8 | dvelimdf.5 | . . . 4 ⊢ (𝜑 → (𝑧 = 𝑦 → (𝜓 ↔ 𝜒))) | |
9 | 1, 7, 8 | sbied 1711 | . . 3 ⊢ (𝜑 → ([𝑦 / 𝑧]𝜓 ↔ 𝜒)) |
10 | 6, 9 | nfbidf 1472 | . 2 ⊢ (𝜑 → (Ⅎ𝑥[𝑦 / 𝑧]𝜓 ↔ Ⅎ𝑥𝜒)) |
11 | 5, 10 | sylibd 147 | 1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 103 ∀wal 1282 Ⅎwnf 1389 [wsb 1685 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 |
This theorem is referenced by: dvelimdc 2238 |
Copyright terms: Public domain | W3C validator |