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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 4281 | . 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐵 ∩ 𝐴) ∖ 𝐶) | |
| 2 | incom 4209 | . 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ 𝐵) | |
| 3 | incom 4209 | . . 3 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
| 4 | 3 | difeq1i 4122 | . 2 ⊢ ((𝐵 ∩ 𝐴) ∖ 𝐶) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
| 5 | 1, 2, 4 | 3eqtr3i 2773 | 1 ⊢ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∖ cdif 3948 ∩ cin 3950 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-in 3958 |
| This theorem is referenced by: resdifcom 6016 resdmdfsn 6049 hartogslem1 9582 fpwwe2 10683 leiso 14498 basdif0 22960 tgdif0 22999 kqdisj 23740 trufil 23918 difininv 32536 gtiso 32710 dfon4 35894 disjdifb 48729 |
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