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Mirrors > Home > MPE Home > Th. List > indifdir | Structured version Visualization version GIF version |
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 4184 | . 2 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐵)) = ((𝐶 ∩ 𝐴) ∖ (𝐶 ∩ 𝐵)) | |
2 | incom 4101 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = (𝐶 ∩ (𝐴 ∖ 𝐵)) | |
3 | incom 4101 | . . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐶 ∩ 𝐴) | |
4 | incom 4101 | . . 3 ⊢ (𝐵 ∩ 𝐶) = (𝐶 ∩ 𝐵) | |
5 | 3, 4 | difeq12i 4021 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶)) = ((𝐶 ∩ 𝐴) ∖ (𝐶 ∩ 𝐵)) |
6 | 1, 2, 5 | 3eqtr4i 2772 | 1 ⊢ ((𝐴 ∖ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∖ (𝐵 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∖ cdif 3850 ∩ cin 3852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2711 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-rab 3063 df-v 3402 df-dif 3856 df-in 3860 |
This theorem is referenced by: resdifdir 6079 preddif 6164 fresaun 6559 uniioombllem4 24350 subsalsal 43480 |
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