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Mirrors > Home > MPE Home > Th. List > indif2 | Structured version Visualization version GIF version |
Description: Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.) |
Ref | Expression |
---|---|
indif2 | ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inass 4196 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (V ∖ 𝐶)) = (𝐴 ∩ (𝐵 ∩ (V ∖ 𝐶))) | |
2 | invdif 4245 | . 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶) | |
3 | invdif 4245 | . . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶) | |
4 | 3 | ineq2i 4186 | . 2 ⊢ (𝐴 ∩ (𝐵 ∩ (V ∖ 𝐶))) = (𝐴 ∩ (𝐵 ∖ 𝐶)) |
5 | 1, 2, 4 | 3eqtr3ri 2853 | 1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3494 ∖ cdif 3933 ∩ cin 3935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-in 3943 |
This theorem is referenced by: indif1 4248 indifcom 4249 wfi 6181 marypha1lem 8897 difopn 21642 restcld 21780 difmbl 24144 voliunlem1 24151 difuncomp 30305 imadifxp 30351 difelcarsg 31568 carsgclctunlem1 31575 frpoind 33080 frind 33085 topbnd 33672 bj-disj2r 34343 nlpineqsn 34692 mblfinlem3 34946 mblfinlem4 34947 rabdif 39127 gneispace 40504 saldifcl2 42631 caragenuncllem 42814 carageniuncllem1 42823 |
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