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Mirrors > Home > MPE Home > Th. List > dvelimdc | Structured version Visualization version GIF version |
Description: Deduction form of dvelimc 3009. Usage of this theorem is discouraged because it depends on ax-13 2389. (Contributed by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.) |
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 1914 | . . 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 2972 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥 𝑤 ∈ 𝐴) |
6 | dvelimdc.4 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑧𝐵) | |
7 | 6 | nfcrd 2972 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑧 𝑤 ∈ 𝐵) |
8 | dvelimdc.5 | . . . . . 6 ⊢ (𝜑 → (𝑧 = 𝑦 → 𝐴 = 𝐵)) | |
9 | eleq2 2904 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵)) | |
10 | 8, 9 | syl6 35 | . . . . 5 ⊢ (𝜑 → (𝑧 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑤 ∈ 𝐵))) |
11 | 2, 3, 5, 7, 10 | dvelimdf 2470 | . . . 4 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑤 ∈ 𝐵)) |
12 | 11 | imp 409 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑤 ∈ 𝐵) |
13 | 1, 12 | nfcd 2971 | . 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐵) |
14 | 13 | ex 415 | 1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 Ⅎwnfc 2964 |
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-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-13 2389 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-cleq 2817 df-clel 2896 df-nfc 2966 |
This theorem is referenced by: dvelimc 3009 |
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