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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 3299 | . . 3 ⊢ (𝐵 ∪ 𝐶) = (𝐶 ∪ 𝐵) | |
2 | 1 | difeq2i 3270 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = (𝐴 ∖ (𝐶 ∪ 𝐵)) |
3 | difun1 3415 | . 2 ⊢ (𝐴 ∖ (𝐵 ∪ 𝐶)) = ((𝐴 ∖ 𝐵) ∖ 𝐶) | |
4 | difun1 3415 | . 2 ⊢ (𝐴 ∖ (𝐶 ∪ 𝐵)) = ((𝐴 ∖ 𝐶) ∖ 𝐵) | |
5 | 2, 3, 4 | 3eqtr3i 2218 | 1 ⊢ ((𝐴 ∖ 𝐵) ∖ 𝐶) = ((𝐴 ∖ 𝐶) ∖ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∖ cdif 3146 ∪ cun 3147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rab 2477 df-v 2758 df-dif 3151 df-un 3153 df-in 3155 |
This theorem is referenced by: difdifdirss 3527 |
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