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Mirrors > Home > MPE Home > Th. List > inindir | Structured version Visualization version GIF version |
Description: Intersection distributes over itself. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
inindir | ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inidm 4235 | . . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶 | |
2 | 1 | ineq2i 4225 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ 𝐶) |
3 | in4 4242 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) | |
4 | 2, 3 | eqtr3i 2765 | 1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∩ cin 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-in 3970 |
This theorem is referenced by: difindir 4299 resindir 6017 predin 6350 restbas 23182 connsuba 23444 kgentopon 23562 trfbas2 23867 trfil2 23911 fclsrest 24048 trust 24254 chtdif 27216 ppidif 27221 mdslmd1lem1 32354 mdslmd1lem2 32355 mddmdin0i 32460 ballotlemgun 34506 cvmsss2 35259 |
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