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| Description: Distribute intersection over difference. (Contributed by Scott Fenton, 14-Apr-2011.) (Revised by BTernaryTau, 14-Aug-2024.) | 
| Ref | Expression | 
|---|---|
| indifdir | ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | indifdi 4294 | . 2 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐵)) = ((𝐶 ∩ 𝐴) ∖ (𝐶 ∩ 𝐵)) | |
| 2 | incom 4209 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∖ 𝐵)) | |
| 3 | incom 4209 | . . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴) | |
| 4 | incom 4209 | . . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵) | |
| 5 | 3, 4 | difeq12i 4124 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶)) = ((𝐶 ∩ 𝐴) ∖ (𝐶 ∩ 𝐵)) | 
| 6 | 1, 2, 5 | 3eqtr4i 2775 | 1 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∖ cdif 3948 ∩ 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-dif 3954 df-in 3958 | 
| This theorem is referenced by: resdifdir 6257 preddif 6350 fresaun 6779 uniioombllem4 25621 subsalsal 46374 | 
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