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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 3325 | . 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐵 ∩ 𝐴) ∖ 𝐶) | |
2 | incom 3273 | . 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ 𝐵) | |
3 | incom 3273 | . . 3 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
4 | 3 | difeq1i 3195 | . 2 ⊢ ((𝐵 ∩ 𝐴) ∖ 𝐶) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
5 | 1, 2, 4 | 3eqtr3i 2169 | 1 ⊢ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∖ cdif 3073 ∩ cin 3075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rab 2426 df-v 2691 df-dif 3078 df-in 3082 |
This theorem is referenced by: resdifcom 4845 resdmdfsn 4870 |
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