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Mirrors > Home > MPE Home > Th. List > djueq2 | Structured version Visualization version 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 2820 | . 2 ⊢ 𝐶 = 𝐶 | |
2 | djueq12 9326 | . 2 ⊢ ((𝐶 = 𝐶 ∧ 𝐴 = 𝐵) → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵)) | |
3 | 1, 2 | mpan 688 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 ⊔ 𝐴) = (𝐶 ⊔ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ⊔ cdju 9320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 df-un 3934 df-opab 5122 df-xp 5554 df-dju 9323 |
This theorem is referenced by: (None) |
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