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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 4236 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) | |
2 | in12 4237 | . 2 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) | |
3 | incom 4217 | . 2 ⊢ (𝐵 ∩ (𝐴 ∩ 𝐶)) = ((𝐴 ∩ 𝐶) ∩ 𝐵) | |
4 | 1, 2, 3 | 3eqtri 2767 | 1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐴 ∩ 𝐶) ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∩ cin 3962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-in 3970 |
This theorem is referenced by: in13 4239 inrot 4241 wefrc 5683 imainrect 6203 sspred 6332 fpwwe2 10681 incexclem 15869 setsfun 17205 setsfun0 17206 ressress 17294 kgeni 23561 kgencn3 23582 fclsrest 24048 voliunlem1 25599 bj-disj2r 37011 refrelsredund4 38614 |
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