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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 3767 | . . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
2 | 1 | ineq1i 3772 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = ((𝐵 ∩ 𝐴) ∩ 𝐶) |
3 | inass 3785 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ (𝐵 ∩ 𝐶)) | |
4 | inass 3785 | . 2 ⊢ ((𝐵 ∩ 𝐴) ∩ 𝐶) = (𝐵 ∩ (𝐴 ∩ 𝐶)) | |
5 | 2, 3, 4 | 3eqtr3i 2640 | 1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = (𝐵 ∩ (𝐴 ∩ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∩ cin 3539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 |
This theorem is referenced by: in32 3787 in31 3789 in4 3791 resdmres 5543 kmlem12 8866 ressress 15765 fh1 27861 fh2 27862 mdslmd3i 28575 bj-inrab3 32117 |
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