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| Description: Class difference by a class difference. (Contributed by Thierry Arnoux, 18-Dec-2017.) | 
| Ref | Expression | 
|---|---|
| difdif2 | ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | difindi 4292 | . 2 ⊢ (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) | |
| 2 | invdif 4279 | . . . 4 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶) | |
| 3 | 2 | eqcomi 2746 | . . 3 ⊢ (𝐵 ∖ 𝐶) = (𝐵 ∩ (V ∖ 𝐶)) | 
| 4 | 3 | difeq2i 4123 | . 2 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = (𝐴 ∖ (𝐵 ∩ (V ∖ 𝐶))) | 
| 5 | dfin2 4271 | . . 3 ⊢ (𝐴 ∩ 𝐶) = (𝐴 ∖ (V ∖ 𝐶)) | |
| 6 | 5 | uneq2i 4165 | . 2 ⊢ ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∖ (V ∖ 𝐶))) | 
| 7 | 1, 4, 6 | 3eqtr4i 2775 | 1 ⊢ (𝐴 ∖ (𝐵 ∖ 𝐶)) = ((𝐴 ∖ 𝐵) ∪ (𝐴 ∩ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 Vcvv 3480 ∖ cdif 3948 ∪ cun 3949 ∩ 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-or 849 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-un 3956 df-in 3958 | 
| This theorem is referenced by: psdmullem 22169 restmetu 24583 difelcarsg 34312 mblfinlem3 37666 mblfinlem4 37667 | 
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