Theorem indif1 3871

Description: Bring an intersection in and out of a class difference. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
indif1	|- ( ( A \ C ) i^i B ) = ( ( A i^i B ) \ C )

Proof of Theorem indif1

Step	Hyp	Ref	Expression
1		indif2 3870	. 2 |- ( B i^i ( A \ C ) ) = ( ( B i^i A ) \ C )
2		incom 3805	. 2 |- ( B i^i ( A \ C ) ) = ( ( A \ C ) i^i B )
3		incom 3805	. . 3 |- ( B i^i A ) = ( A i^i B )
4	3	difeq1i 3724	. 2 |- ( ( B i^i A ) \ C ) = ( ( A i^i B ) \ C )
5	1, 2, 4	3eqtr3i 2652	1 |- ( ( A \ C ) i^i B ) = ( ( A i^i B ) \ C )

Syntax hints:   = wceq 1483   \ 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:  resdifcom  5415  resdmdfsn  5445  hartogslem1  8447  fpwwe2  9465  leiso  13243  basdif0  20757  tgdif0  20796  kqdisj  21535  trufil  21714  difininv  29354  gtiso  29478  dfon4  32000