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Mirrors > Home > ILE Home > Th. List > dvelimf | GIF version |
Description: Version of dvelim 1994 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 1989 | . 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 1767 | . 2 ⊢ ([𝑦 / 𝑧]𝜑 ↔ 𝜓) |
6 | 5 | albii 1447 | . 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 1330 [wsb 1739 |
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 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1740 |
This theorem is referenced by: dvelim 1994 dveel1 2134 dveel2 2135 |
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