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Mirrors > Home > ILE Home > Th. List > dif32 | GIF version |
Description: Swap second and third argument of double difference. (Contributed by NM, 18-Aug-2004.) |
Ref | Expression |
---|---|
dif32 | ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3087 | . . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
2 | 1 | difeq2i 3059 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = (𝐴 ∖ (𝐶 ∪ 𝐵)) |
3 | difun1 3197 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∖ 𝐶) | |
4 | difun1 3197 | . 2 ⊢ (𝐴 ∖ (𝐶 ∪ 𝐵)) = ((𝐴 ∖ 𝐶) ∖ 𝐵) | |
5 | 2, 3, 4 | 3eqtr3i 2068 | 1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∖ cdif 2914 ∪ cun 2915 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 |
This theorem is referenced by: difdifdirss 3307 |
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