Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dvelimf | GIF version |
Description: Version of dvelim 1970 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 1965 | . 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 1748 | . 2 ⊢ ([𝑦 / 𝑧]𝜑 ↔ 𝜓) |
6 | 5 | albii 1431 | . 2 ⊢ (∀𝑥[𝑦 / 𝑧]𝜑 ↔ ∀𝑥𝜓) |
7 | 2, 5, 6 | 3imtr3g 203 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∀wal 1314 [wsb 1720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 |
This theorem is referenced by: dvelim 1970 dveel1 1973 dveel2 1974 |
Copyright terms: Public domain | W3C validator |