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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 4194 | . . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶 | |
2 | 1 | ineq2i 4185 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ 𝐶) |
3 | in4 4201 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) | |
4 | 2, 3 | eqtr3i 2846 | 1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∩ cin 3934 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3942 |
This theorem is referenced by: difindir 4258 resindir 5864 predin 6165 restbas 21760 connsuba 22022 kgentopon 22140 trfbas2 22445 trfil2 22489 fclsrest 22626 trust 22832 chtdif 25729 ppidif 25734 mdslmd1lem1 30096 mdslmd1lem2 30097 mddmdin0i 30202 ballotlemgun 31777 cvmsss2 32516 |
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