![]() |
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 1730 | . 2 ⊢ Ⅎ𝑧∀𝑥 𝑥 = 𝑦 | |
2 | a16g 1875 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑧𝜑)) | |
3 | 1, 2 | nfd 1534 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → Ⅎ𝑧𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1362 Ⅎwnf 1471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 |
This theorem is referenced by: nfsbxy 1954 nfsbxyt 1955 dvelimor 2030 |
Copyright terms: Public domain | W3C validator |