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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 3822 | . . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶 | |
2 | 1 | ineq2i 3811 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ 𝐶) |
3 | in4 3829 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) | |
4 | 2, 3 | eqtr3i 2646 | 1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∩ cin 3573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 |
This theorem is referenced by: difindir 3882 resindir 5413 predin 5703 restbas 20962 connsuba 21223 kgentopon 21341 trfbas2 21647 trfil2 21691 fclsrest 21828 trust 22033 chtdif 24884 ppidif 24889 mdslmd1lem1 29184 mdslmd1lem2 29185 mddmdin0i 29290 ballotlemgun 30586 cvmsss2 31256 |
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