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Mirrors > Home > NFE 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 3466 | . . 3 ⊢ (B ∩ (C ∩ D)) = (C ∩ (B ∩ D)) | |
2 | 1 | ineq2i 3454 | . 2 ⊢ (A ∩ (B ∩ (C ∩ D))) = (A ∩ (C ∩ (B ∩ D))) |
3 | inass 3465 | . 2 ⊢ ((A ∩ B) ∩ (C ∩ D)) = (A ∩ (B ∩ (C ∩ D))) | |
4 | inass 3465 | . 2 ⊢ ((A ∩ C) ∩ (B ∩ D)) = (A ∩ (C ∩ (B ∩ D))) | |
5 | 2, 3, 4 | 3eqtr4i 2383 | 1 ⊢ ((A ∩ B) ∩ (C ∩ D)) = ((A ∩ C) ∩ (B ∩ D)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∩ cin 3208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 |
This theorem is referenced by: inindi 3472 inindir 3473 |
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