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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 4156 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐶)) | |
2 | uneq2 4157 | . 2 ⊢ (𝐶 = 𝐷 → (𝐵 ∪ 𝐶) = (𝐵 ∪ 𝐷)) | |
3 | 1, 2 | sylan9eq 2793 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∪ cun 3946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-un 3953 |
This theorem is referenced by: uneq12i 4161 uneq12d 4164 un00 4442 opthprc 5739 dmpropg 6212 unixp 6279 fntpg 6606 fnun 6661 resasplit 6759 fvun 6979 rankprb 9843 pm54.43 9993 pwmndgplus 18813 evlseu 21638 ptuncnv 23303 sshjval 30591 bj-2upleq 35882 bj-unexg 35908 poimirlem4 36481 poimirlem9 36486 evlselvlem 41156 diophun 41497 pwssplit4 41817 clsk1indlem3 42780 |
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