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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 3116 | . . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
2 | 1 | difeq2i 3087 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = (𝐴 ∖ (𝐶 ∪ 𝐵)) |
3 | difun1 3224 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∖ 𝐶) | |
4 | difun1 3224 | . 2 ⊢ (𝐴 ∖ (𝐶 ∪ 𝐵)) = ((𝐴 ∖ 𝐶) ∖ 𝐵) | |
5 | 2, 3, 4 | 3eqtr3i 2109 | 1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∖ cdif 2970 ∪ cun 2971 |
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-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rab 2357 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 |
This theorem is referenced by: difdifdirss 3327 |
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