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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 4145 | . . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶 | |
2 | 1 | ineq2i 4136 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ 𝐶) |
3 | in4 4152 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) | |
4 | 2, 3 | eqtr3i 2823 | 1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∩ cin 3880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-in 3888 |
This theorem is referenced by: difindir 4209 resindir 5835 predin 6139 restbas 21763 connsuba 22025 kgentopon 22143 trfbas2 22448 trfil2 22492 fclsrest 22629 trust 22835 chtdif 25743 ppidif 25748 mdslmd1lem1 30108 mdslmd1lem2 30109 mddmdin0i 30214 ballotlemgun 31892 cvmsss2 32634 |
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