Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > a16nf | GIF version |
Description: If there is only one element in the universe, then everything satisfies Ⅎ. (Contributed by Mario Carneiro, 7-Oct-2016.) |
Ref | Expression |
---|---|
a16nf | ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfae 1647 | . 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦 | |
2 | a16g 1785 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | |
3 | 1, 2 | nfd 1456 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 Ⅎwnf 1389 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 |
This theorem is referenced by: nfsbxy 1859 nfsbxyt 1860 dvelimor 1935 |
Copyright terms: Public domain | W3C validator |