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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 4271 | . 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐵 ∩ 𝐴) ∖ 𝐶) | |
2 | incom 4202 | . 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐴 ∖ 𝐶) ∩ 𝐵) | |
3 | incom 4202 | . . 3 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵) | |
4 | 3 | difeq1i 4119 | . 2 ⊢ ((𝐵 ∩ 𝐴) ∖ 𝐶) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
5 | 1, 2, 4 | 3eqtr3i 2766 | 1 ⊢ ((𝐴 ∖ 𝐶) ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∖ cdif 3946 ∩ cin 3948 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 df-dif 3952 df-in 3956 |
This theorem is referenced by: resdifcom 6001 resdmdfsn 6032 hartogslem1 9541 fpwwe2 10642 leiso 14426 basdif0 22678 tgdif0 22717 kqdisj 23458 trufil 23636 difininv 32020 gtiso 32187 dfon4 35167 disjdifb 47583 |
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