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Mirrors > Home > MPE Home > Th. List > inrot | Structured version Visualization version GIF version |
Description: Rotate the intersection of 3 classes. (Contributed by NM, 27-Aug-2012.) |
Ref | Expression |
---|---|
inrot | ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | in31 4224 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐵) ∩ 𝐴) | |
2 | in32 4222 | . 2 ⊢ ((𝐶 ∩ 𝐵) ∩ 𝐴) = ((𝐶 ∩ 𝐴) ∩ 𝐵) | |
3 | 1, 2 | eqtri 2756 | 1 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐶 ∩ 𝐴) ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∩ cin 3946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-in 3954 |
This theorem is referenced by: (None) |
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