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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 4219 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
2 | 1 | ineq1i 4209 | . 2 ⊢ ((𝐴 ∩ 𝐴) ∩ (𝐵 ∩ 𝐶)) = (𝐴 ∩ (𝐵 ∩ 𝐶)) |
3 | in4 4226 | . 2 ⊢ ((𝐴 ∩ 𝐴) ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) | |
4 | 2, 3 | eqtr3i 2763 | 1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∩ cin 3948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-in 3956 |
This theorem is referenced by: difundi 4280 dfif5 4545 resindi 5998 offres 7970 incexclem 15782 bitsinv1 16383 bitsinvp1 16390 bitsres 16414 fh1 30871 |
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