![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > in4 | GIF version |
Description: Rearrangement of intersection of 4 classes. (Contributed by NM, 21-Apr-2001.) |
Ref | Expression |
---|---|
in4 | ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | in12 3197 | . . 3 ⊢ (𝐵 ∩ (𝐶 ∩ 𝐷)) = (𝐶 ∩ (𝐵 ∩ 𝐷)) | |
2 | 1 | ineq2i 3184 | . 2 ⊢ (𝐴 ∩ (𝐵 ∩ (𝐶 ∩ 𝐷))) = (𝐴 ∩ (𝐶 ∩ (𝐵 ∩ 𝐷))) |
3 | inass 3196 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = (𝐴 ∩ (𝐵 ∩ (𝐶 ∩ 𝐷))) | |
4 | inass 3196 | . 2 ⊢ ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷)) = (𝐴 ∩ (𝐶 ∩ (𝐵 ∩ 𝐷))) | |
5 | 2, 3, 4 | 3eqtr4i 2115 | 1 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐶 ∩ 𝐷)) = ((𝐴 ∩ 𝐶) ∩ (𝐵 ∩ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1287 ∩ cin 2985 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 |
This theorem depends on definitions: df-bi 115 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-v 2616 df-in 2992 |
This theorem is referenced by: inindi 3203 inindir 3204 |
Copyright terms: Public domain | W3C validator |