Theorem indif2 4228

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 4175	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (V ∖ 𝐶)) = (𝐴 ∩ (𝐵 ∩ (V ∖ 𝐶)))
2		invdif 4226	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
3		invdif 4226	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
4	3	ineq2i 4164	. 2 ⊢ (𝐴 ∩ (𝐵 ∩ (V ∖ 𝐶))) = (𝐴 ∩ (𝐵 ∖ 𝐶))
5	1, 2, 4	3eqtr3ri 2763	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)

Syntax hints:   = wceq 1541  Vcvv 3436   ∖ cdif 3894   ∩ cin 3896

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-in 3904

This theorem is referenced by:  indif1  4229  indifcom  4230  rabdif  4268  frpoind  6289  marypha1lem  9317  frind  9643  difopn  22949  restcld  23087  difmbl  25471  voliunlem1  25478  difuncomp  32533  imadifxp  32581  psrbasfsupp  33572  difelcarsg  34323  carsgclctunlem1  34330  topbnd  36368  bj-disj2r  37072  nlpineqsn  37452  mblfinlem3  37698  mblfinlem4  37699  dmxrncnvepres2  38456  gneispace  44226  saldifcl2  46425  caragenuncllem  46609  carageniuncllem1  46618  iscnrm3rlem1  49039