Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > inindi | Structured version Visualization version GIF version |
Description: Intersection distributes over itself. (Contributed by NM, 6-May-1994.) |
Ref | Expression |
---|---|
inindi | ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inidm 3784 | . . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴 | |
2 | 1 | ineq1i 3772 | . 2 ⊢ ((𝐴 ∩ 𝐴) ∩ (𝐵 ∩ 𝐶)) = (𝐴 ∩ (𝐵 ∩ 𝐶)) |
3 | in4 3791 | . 2 ⊢ ((𝐴 ∩ 𝐴) ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶)) | |
4 | 2, 3 | eqtr3i 2634 | 1 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐶)) = ((𝐴 ∩ 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: difundi 3838 dfif5 4052 resindi 5332 offres 7054 incexclem 14407 bitsinv1 15002 bitsinvp1 15009 bitsres 15033 fh1 27861 |
Copyright terms: Public domain | W3C validator |