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Mirrors > Home > MPE Home > Th. List > in32 | Structured version Visualization version GIF version |
Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
in32 | ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inass 4159 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) | |
2 | in12 4160 | . 2 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) | |
3 | incom 4140 | . 2 ⊢ (𝐵 ∩ (𝐴 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ 𝐵) | |
4 | 1, 2, 3 | 3eqtri 2772 | 1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∩ cin 3891 |
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 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1545 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-rab 3075 df-v 3433 df-in 3899 |
This theorem is referenced by: in13 4162 inrot 4164 wefrc 5583 imainrect 6082 sspred 6209 fpwwe2 10392 incexclem 15538 setsfun 16862 setsfun0 16863 ressress 16948 kgeni 22678 kgencn3 22699 fclsrest 23165 voliunlem1 24704 bj-disj2r 35206 refrelsredund4 36733 |
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