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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 2795 | . 2 ⊢ 𝐶 = 𝐶 | |
2 | djueq12 9179 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) | |
3 | 1, 2 | mpan2 687 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ⊔ cdju 9173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-v 3439 df-un 3864 df-opab 5025 df-xp 5449 df-dju 9176 |
This theorem is referenced by: djulepw 9464 |
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