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Mirrors > Home > MPE Home > Th. List > in12 | Structured version Visualization version GIF version |
Description: A rearrangement of intersection. (Contributed by NM, 21-Apr-2001.) |
Ref | Expression |
---|---|
in12 | ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4128 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | 1 | ineq1i 4135 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐵 ∩ 𝐴) ∩ 𝐶) |
3 | inass 4146 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) | |
4 | inass 4146 | . 2 ⊢ ((𝐵 ∩ 𝐴) ∩ 𝐶) = (𝐵 ∩ (𝐴 ∩ 𝐶)) | |
5 | 2, 3, 4 | 3eqtr3i 2829 | 1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∩ cin 3880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-in 3888 |
This theorem is referenced by: in32 4148 in31 4150 in4 4152 resdmres 6056 kmlem12 9572 ressress 16554 fh1 29401 fh2 29402 mdslmd3i 30115 bj-inrab3 34371 |
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