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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 2726 | . 2 ⊢ 𝐶 = 𝐶 | |
2 | djueq12 9940 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) | |
3 | 1, 2 | mpan2 689 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ⊔ cdju 9934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-un 3951 df-opab 5208 df-xp 5680 df-dju 9937 |
This theorem is referenced by: djulepw 10228 |
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