Theorem indif2 4287

Description: Bring an intersection in and out of a class difference. (Contributed by Jeff Hankins, 15-Jul-2009.)

Assertion

Ref	Expression
indif2	⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)

Proof of Theorem indif2

Step	Hyp	Ref	Expression
1		inass 4236	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (V ∖ 𝐶)) = (𝐴 ∩ (𝐵 ∩ (V ∖ 𝐶)))
2		invdif 4285	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
3		invdif 4285	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
4	3	ineq2i 4225	. 2 ⊢ (𝐴 ∩ (𝐵 ∩ (V ∖ 𝐶))) = (𝐴 ∩ (𝐵 ∖ 𝐶))
5	1, 2, 4	3eqtr3ri 2772	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)

Syntax hints:   = wceq 1537  Vcvv 3478   ∖ cdif 3960   ∩ cin 3962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-in 3970

This theorem is referenced by:  indif1  4288  indifcom  4289  rabdif  4327  frpoind  6365  wfiOLD  6374  marypha1lem  9471  frind  9788  difopn  23058  restcld  23196  difmbl  25592  voliunlem1  25599  difuncomp  32574  imadifxp  32621  difelcarsg  34292  carsgclctunlem1  34299  topbnd  36307  bj-disj2r  37011  nlpineqsn  37391  mblfinlem3  37646  mblfinlem4  37647  gneispace  44124  saldifcl2  46284  caragenuncllem  46468  carageniuncllem1  46477  iscnrm3rlem1  48737