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Mirrors > Home > ILE Home > Th. List > dvelimf | GIF version |
Description: Version of dvelim 1893 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) |
Ref | Expression |
---|---|
dvelimf.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
dvelimf.2 | ⊢ (𝜓 → ∀𝑧𝜓) |
dvelimf.3 | ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
dvelimf | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvelimf.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | 1 | hbsb4 1888 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → ∀𝑥[𝑦 / 𝑧]𝜑)) |
3 | dvelimf.2 | . . 3 ⊢ (𝜓 → ∀𝑧𝜓) | |
4 | dvelimf.3 | . . 3 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbieh 1673 | . 2 ⊢ ([𝑦 / 𝑧]𝜑 ↔ 𝜓) |
6 | 5 | albii 1359 | . 2 ⊢ (∀𝑥[𝑦 / 𝑧]𝜑 ↔ ∀𝑥𝜓) |
7 | 2, 5, 6 | 3imtr3g 193 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ∀wal 1241 [wsb 1645 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 |
This theorem is referenced by: dvelim 1893 dveel1 1896 dveel2 1897 |
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