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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 4227 | . . 3 ⊢ (𝐶 ∩ 𝐶) = 𝐶 | |
| 2 | 1 | ineq2i 4217 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ 𝐶) | 
| 3 | in4 4234 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) | |
| 4 | 2, 3 | eqtr3i 2767 | 1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∩ cin 3950 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-in 3958 | 
| This theorem is referenced by: difindir 4293 resindir 6014 predin 6348 restbas 23166 connsuba 23428 kgentopon 23546 trfbas2 23851 trfil2 23895 fclsrest 24032 trust 24238 chtdif 27201 ppidif 27206 mdslmd1lem1 32344 mdslmd1lem2 32345 mddmdin0i 32450 ballotlemgun 34527 cvmsss2 35279 | 
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