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Mirrors > Home > MPE Home > Th. List > dvelimdf | Structured version Visualization version GIF version |
Description: Deduction form of dvelimf 2449. Usage of this theorem is discouraged because it depends on ax-13 2373. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (New usage is discouraged.) |
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.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | dvelimdf.3 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
3 | 1, 2 | nfim1 2201 | . . 3 ⊢ Ⅎ𝑥(𝜑 → 𝜓) |
4 | dvelimdf.2 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
5 | dvelimdf.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑧𝜒) | |
6 | 4, 5 | nfim1 2201 | . . 3 ⊢ Ⅎ𝑧(𝜑 → 𝜒) |
7 | dvelimdf.5 | . . . . 5 ⊢ (𝜑 → (𝑧 = 𝑦 → (𝜓 ↔ 𝜒))) | |
8 | 7 | com12 32 | . . . 4 ⊢ (𝑧 = 𝑦 → (𝜑 → (𝜓 ↔ 𝜒))) |
9 | 8 | pm5.74d 276 | . . 3 ⊢ (𝑧 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
10 | 3, 6, 9 | dvelimf 2449 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝜑 → 𝜒)) |
11 | pm5.5 365 | . . 3 ⊢ (𝜑 → ((𝜑 → 𝜒) ↔ 𝜒)) | |
12 | 1, 11 | nfbidf 2226 | . 2 ⊢ (𝜑 → (Ⅎ𝑥(𝜑 → 𝜒) ↔ Ⅎ𝑥𝜒)) |
13 | 10, 12 | syl5ib 247 | 1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∀wal 1540 Ⅎwnf 1790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-11 2162 ax-12 2179 ax-13 2373 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 |
This theorem is referenced by: nfsb4t 2504 dvelimdc 2927 |
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