Theorem indif2 3870

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

Assertion

Ref	Expression
indif2	|- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ C )

Proof of Theorem indif2

Step	Hyp	Ref	Expression
1		inass 3823	. 2 |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( A i^i ( B i^i ( _V \ C ) ) )
2		invdif 3868	. 2 |- ( ( A i^i B ) i^i ( _V \ C ) ) = ( ( A i^i B ) \ C )
3		invdif 3868	. . 3 |- ( B i^i ( _V \ C ) ) = ( B \ C )
4	3	ineq2i 3811	. 2 |- ( A i^i ( B i^i ( _V \ C ) ) ) = ( A i^i ( B \ C ) )
5	1, 2, 4	3eqtr3ri 2653	1 |- ( A i^i ( B \ C ) ) = ( ( A i^i B ) \ C )

Syntax hints:   = wceq 1483   _Vcvv 3200   \ cdif 3571   i^i cin 3573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581

This theorem is referenced by:  indif1  3871  indifcom  3872  wfi  5713  marypha1lem  8339  difopn  20838  restcld  20976  difmbl  23311  voliunlem1  23318  difuncomp  29369  imadifxp  29414  difelcarsg  30372  carsgclctunlem1  30379  frind  31740  topbnd  32319  bj-disj2r  33013  mblfinlem3  33448  mblfinlem4  33449  gneispace  38432  saldifcl2  40546  caragenuncllem  40726  carageniuncllem1  40735