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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 1654 | . 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦 | |
2 | a16g 1792 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | |
3 | 1, 2 | nfd 1461 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1287 Ⅎwnf 1394 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 |
This theorem is referenced by: nfsbxy 1866 nfsbxyt 1867 dvelimor 1942 |
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