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Mirrors > Home > MPE Home > Th. List > inindi | Structured version Visualization version GIF version |
Description: Intersection distributes over itself. (Contributed by NM, 6-May-1994.) |
Ref | Expression |
---|---|
inindi | ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inidm 4197 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
2 | 1 | ineq1i 4187 | . 2 ⊢ ((𝐴 ∩ 𝐴) ∩ (𝐵 ∩ 𝐶)) = (𝐴 ∩ (𝐵 ∩ 𝐶)) |
3 | in4 4204 | . 2 ⊢ ((𝐴 ∩ 𝐴) ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) | |
4 | 2, 3 | eqtr3i 2848 | 1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∩ cin 3937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-in 3945 |
This theorem is referenced by: difundi 4258 dfif5 4485 resindi 5871 offres 7686 incexclem 15193 bitsinv1 15793 bitsinvp1 15800 bitsres 15824 fh1 29397 |
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