Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > djueq2 | GIF version |
Description: Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
Ref | Expression |
---|---|
djueq2 | ⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2137 | . 2 ⊢ 𝐶 = 𝐶 | |
2 | djueq12 6917 | . 2 ⊢ ((𝐶 = 𝐶 ∧ 𝐴 = 𝐵) → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵)) | |
3 | 1, 2 | mpan 420 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ⊔ cdju 6915 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-opab 3985 df-xp 4540 df-dju 6916 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |