Theorem difeq2 3316

Description: Equality theorem for class difference. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
difeq2	|- ( A = B -> ( C \ A ) = ( C \ B ) )

Proof of Theorem difeq2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2293	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	notbid 671	. . 3 |- ( A = B -> ( -. x e. A <-> -. x e. B ) )
3	2	rabbidv 2788	. 2 |- ( A = B -> { x e. C | -. x e. A } = { x e. C | -. x e. B } )
4		dfdif2 3205	. 2 |- ( C \ A ) = { x e. C | -. x e. A }
5		dfdif2 3205	. 2 |- ( C \ B ) = { x e. C | -. x e. B }
6	3, 4, 5	3eqtr4g 2287	1 |- ( A = B -> ( C \ A ) = ( C \ B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1395   e. wcel 2200   {crab 2512   \ cdif 3194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ral 2513  df-rab 2517  df-dif 3199

This theorem is referenced by:  difeq12  3317  difeq2i  3319  difeq2d  3322  disjdif2  3570  ssdifeq0  3574  2oconcl  6585  diffitest  7049  diffifi  7056  undifdc  7086  sbthlem2  7125  isbth  7134  difinfinf  7268  ismkvnex  7322  iscld  14777