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Mirrors > Home > MPE Home > Th. List > uneq12 | Structured version Visualization version GIF version |
Description: Equality theorem for the union of two classes. (Contributed by NM, 29-Mar-1998.) |
Ref | Expression |
---|---|
uneq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1 4129 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) | |
2 | uneq2 4130 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | |
3 | 1, 2 | sylan9eq 2873 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∪ cun 3931 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-un 3938 |
This theorem is referenced by: uneq12i 4134 uneq12d 4137 un00 4390 opthprc 5609 dmpropg 6065 unixp 6126 fntpg 6407 fnun 6456 resasplit 6541 fvun 6746 rankprb 9268 pm54.43 9417 pwmndgplus 18038 evlseu 20224 ptuncnv 22343 sshjval 29054 bj-2upleq 34221 poimirlem4 34777 poimirlem9 34782 diophun 39248 pwssplit4 39567 clsk1indlem3 40271 |
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