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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 4218 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) | |
2 | in12 4219 | . 2 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) | |
3 | incom 4199 | . 2 ⊢ (𝐵 ∩ (𝐴 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ 𝐵) | |
4 | 1, 2, 3 | 3eqtri 2757 | 1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∩ cin 3943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-in 3951 |
This theorem is referenced by: in13 4221 inrot 4223 wefrc 5672 imainrect 6187 sspred 6316 fpwwe2 10673 incexclem 15826 setsfun 17159 setsfun0 17160 ressress 17248 kgeni 23502 kgencn3 23523 fclsrest 23989 voliunlem1 25540 bj-disj2r 36658 refrelsredund4 38254 |
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