Theorem difdif2 3884

Description: Class difference by a class difference. (Contributed by Thierry Arnoux, 18-Dec-2017.)

Assertion

Ref	Expression
difdif2	|- ( A \ ( B \ C ) ) = ( ( A \ B ) u. ( A i^i C ) )

Proof of Theorem difdif2

Step	Hyp	Ref	Expression
1		difindi 3881	. 2 |- ( A \ ( B i^i ( _V \ C ) ) ) = ( ( A \ B ) u. ( A \ ( _V \ C ) ) )
2		invdif 3868	. . . 4 |- ( B i^i ( _V \ C ) ) = ( B \ C )
3	2	eqcomi 2631	. . 3 |- ( B \ C ) = ( B i^i ( _V \ C ) )
4	3	difeq2i 3725	. 2 |- ( A \ ( B \ C ) ) = ( A \ ( B i^i ( _V \ C ) ) )
5		dfin2 3860	. . 3 |- ( A i^i C ) = ( A \ ( _V \ C ) )
6	5	uneq2i 3764	. 2 |- ( ( A \ B ) u. ( A i^i C ) ) = ( ( A \ B ) u. ( A \ ( _V \ C ) ) )
7	1, 4, 6	3eqtr4i 2654	1 |- ( A \ ( B \ C ) ) = ( ( A \ B ) u. ( A i^i C ) )

Syntax hints:   = wceq 1483   _Vcvv 3200   \ cdif 3571   u. cun 3572   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-un 3579  df-in 3581

This theorem is referenced by:  restmetu  22375  difelcarsg  30372  mblfinlem3  33448  mblfinlem4  33449