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Mirrors > Home > MPE Home > Th. List > dvelimdf | Structured version Visualization version GIF version |
Description: Deduction form of dvelimf 2448. Usage of this theorem is discouraged because it depends on ax-13 2372. (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 2192 | . . 3 ⊢ Ⅎ𝑥(𝜑 → 𝜓) |
4 | dvelimdf.2 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
5 | dvelimdf.4 | . . . 4 ⊢ (𝜑 → Ⅎ𝑧𝜒) | |
6 | 4, 5 | nfim1 2192 | . . 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 272 | . . 3 ⊢ (𝑧 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
10 | 3, 6, 9 | dvelimf 2448 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝜑 → 𝜒)) |
11 | pm5.5 362 | . . 3 ⊢ (𝜑 → ((𝜑 → 𝜒) ↔ 𝜒)) | |
12 | 1, 11 | nfbidf 2217 | . 2 ⊢ (𝜑 → (Ⅎ𝑥(𝜑 → 𝜒) ↔ Ⅎ𝑥𝜒)) |
13 | 10, 12 | syl5ib 243 | 1 ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∀wal 1537 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 1913 ax-6 1971 ax-7 2011 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 |
This theorem is referenced by: nfsb4t 2503 dvelimdc 2934 |
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