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Mirrors > Home > ILE Home > Th. List > dvelimdc | GIF version |
Description: Deduction form of dvelimc 2239. (Contributed by Mario Carneiro, 8-Oct-2016.) |
Ref | Expression |
---|---|
dvelimdc.1 | ⊢ Ⅎ𝑥𝜑 |
dvelimdc.2 | ⊢ Ⅎ𝑧𝜑 |
dvelimdc.3 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
dvelimdc.4 | ⊢ (𝜑 → Ⅎ𝑧𝐵) |
dvelimdc.5 | ⊢ (𝜑 → (𝑧 = 𝑦 → 𝐴 = 𝐵)) |
Ref | Expression |
---|---|
dvelimdc | ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1461 | . . 3 ⊢ Ⅎ𝑤(𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) | |
2 | dvelimdc.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
3 | dvelimdc.2 | . . . . 5 ⊢ Ⅎ𝑧𝜑 | |
4 | dvelimdc.3 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | 4 | nfcrd 2232 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑤 ∈ 𝐴) |
6 | dvelimdc.4 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝐵) | |
7 | 6 | nfcrd 2232 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑧 𝑤 ∈ 𝐵) |
8 | dvelimdc.5 | . . . . . 6 ⊢ (𝜑 → (𝑧 = 𝑦 → 𝐴 = 𝐵)) | |
9 | eleq2 2142 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵)) | |
10 | 8, 9 | syl6 33 | . . . . 5 ⊢ (𝜑 → (𝑧 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))) |
11 | 2, 3, 5, 7, 10 | dvelimdf 1933 | . . . 4 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤 ∈ 𝐵)) |
12 | 11 | imp 122 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑤 ∈ 𝐵) |
13 | 1, 12 | nfcd 2214 | . 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐵) |
14 | 13 | ex 113 | 1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1282 = wceq 1284 Ⅎwnf 1389 ∈ wcel 1433 Ⅎwnfc 2206 |
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 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-cleq 2074 df-clel 2077 df-nfc 2208 |
This theorem is referenced by: dvelimc 2239 |
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