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Mirrors > Home > NFE Home > Th. List > undir | GIF version |
Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.) |
Ref | Expression |
---|---|
undir | ⊢ ((A ∩ B) ∪ C) = ((A ∪ C) ∩ (B ∪ C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | undi 3502 | . 2 ⊢ (C ∪ (A ∩ B)) = ((C ∪ A) ∩ (C ∪ B)) | |
2 | uncom 3408 | . 2 ⊢ ((A ∩ B) ∪ C) = (C ∪ (A ∩ B)) | |
3 | uncom 3408 | . . 3 ⊢ (A ∪ C) = (C ∪ A) | |
4 | uncom 3408 | . . 3 ⊢ (B ∪ C) = (C ∪ B) | |
5 | 3, 4 | ineq12i 3455 | . 2 ⊢ ((A ∪ C) ∩ (B ∪ C)) = ((C ∪ A) ∩ (C ∪ B)) |
6 | 1, 2, 5 | 3eqtr4i 2383 | 1 ⊢ ((A ∩ B) ∪ C) = ((A ∪ C) ∩ (B ∪ C)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∪ cun 3207 ∩ cin 3208 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 df-un 3214 |
This theorem is referenced by: undif1 3625 dfif4 3673 dfif5 3674 nnsucelrlem3 4426 |
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