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Mirrors > Home > ILE Home > Th. List > indif1 | GIF version |
Description: Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
indif1 | ⊢ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indif2 3208 | . 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐵 ∩ 𝐴) ∖ 𝐶) | |
2 | incom 3158 | . 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ 𝐵) | |
3 | incom 3158 | . . 3 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
4 | 3 | difeq1i 3086 | . 2 ⊢ ((𝐵 ∩ 𝐴) ∖ 𝐶) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
5 | 1, 2, 4 | 3eqtr3i 2109 | 1 ⊢ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∖ cdif 2970 ∩ cin 2972 |
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-rab 2357 df-v 2603 df-dif 2975 df-in 2979 |
This theorem is referenced by: resdmdfsn 4671 |
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