Theorem indif2 4256

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 4203	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (V ∖ 𝐶)) = (𝐴 ∩ (𝐵 ∩ (V ∖ 𝐶)))
2		invdif 4254	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
3		invdif 4254	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
4	3	ineq2i 4192	. 2 ⊢ (𝐴 ∩ (𝐵 ∩ (V ∖ 𝐶))) = (𝐴 ∩ (𝐵 ∖ 𝐶))
5	1, 2, 4	3eqtr3ri 2767	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)

Syntax hints:   = wceq 1540  Vcvv 3459   ∖ cdif 3923   ∩ cin 3925

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 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-in 3933

This theorem is referenced by:  indif1  4257  indifcom  4258  rabdif  4296  frpoind  6331  wfiOLD  6340  marypha1lem  9445  frind  9764  difopn  22972  restcld  23110  difmbl  25496  voliunlem1  25503  difuncomp  32534  imadifxp  32582  difelcarsg  34342  carsgclctunlem1  34349  topbnd  36342  bj-disj2r  37046  nlpineqsn  37426  mblfinlem3  37683  mblfinlem4  37684  gneispace  44158  saldifcl2  46357  caragenuncllem  46541  carageniuncllem1  46550  iscnrm3rlem1  48914