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Mirrors > Home > MPE Home > Th. List > djueq1 | Structured version Visualization version GIF version |
Description: Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
Ref | Expression |
---|---|
djueq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2737 | . 2 ⊢ 𝐶 = 𝐶 | |
2 | djueq12 9520 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) | |
3 | 1, 2 | mpan2 691 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ⊔ cdju 9514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-un 3871 df-opab 5116 df-xp 5557 df-dju 9517 |
This theorem is referenced by: djulepw 9806 |
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