Theorem indif2 4233

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 4179	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (V ∖ 𝐶)) = (𝐴 ∩ (𝐵 ∩ (V ∖ 𝐶)))
2		invdif 4231	. 2 ⊢ ((𝐴 ∩ 𝐵) ∩ (V ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
3		invdif 4231	. . 3 ⊢ (𝐵 ∩ (V ∖ 𝐶)) = (𝐵 ∖ 𝐶)
4	3	ineq2i 4169	. 2 ⊢ (𝐴 ∩ (𝐵 ∩ (V ∖ 𝐶))) = (𝐴 ∩ (𝐵 ∖ 𝐶))
5	1, 2, 4	3eqtr3ri 2794	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)

This proof depends on syntax axioms:   = wceq 1569  Vcvv 3454   ∖ cdif 3901   ∩ cin 3903

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-dif 3907  df-in 3911

This theorem is used by:  indif1  4234  indifcom  4235  rabdif  4273  inssdif0  4328  frpoind  6343  marypha1lem  9391  frind  9720  difopn  23202  restcld  23340  difmbl  25713  voliunlem1  25720  difuncomp  32909  imadifxp  32957  psrbasfsupp  33910  difelcarsg  34709  carsgclctunlem1  34716  topbnd  36863  bj-disj2r  37692  nlpineqsn  38082  mblfinlem3  38338  mblfinlem4  38339  dmxrncnvepres2  39110  gneispace  44888  saldifcl2  47070  caragenuncllem  47254  carageniuncllem1  47263  iscnrm3rlem1  49746