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Mirrors > Home > MPE Home > Th. List > indif1 | Structured version Visualization version 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 3829 | . 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐵 ∩ 𝐴) ∖ 𝐶) | |
2 | incom 3767 | . 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ 𝐵) | |
3 | incom 3767 | . . 3 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
4 | 3 | difeq1i 3686 | . 2 ⊢ ((𝐵 ∩ 𝐴) ∖ 𝐶) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
5 | 1, 2, 4 | 3eqtr3i 2640 | 1 ⊢ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∖ cdif 3537 ∩ cin 3539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-in 3547 |
This theorem is referenced by: resdifcom 5335 resdmdfsn 5365 hartogslem1 8330 fpwwe2 9344 leiso 13100 basdif0 20568 tgdif0 20607 kqdisj 21345 trufil 21524 gtiso 28861 dfon4 31170 |
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