Theorem difeq2 3293

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 2271	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	notbid 669	. . 3 |- ( A = B -> ( -. x e. A <-> -. x e. B ) )
3	2	rabbidv 2765	. 2 |- ( A = B -> { x e. C | -. x e. A } = { x e. C | -. x e. B } )
4		dfdif2 3182	. 2 |- ( C \ A ) = { x e. C | -. x e. A }
5		dfdif2 3182	. 2 |- ( C \ B ) = { x e. C | -. x e. B }
6	3, 4, 5	3eqtr4g 2265	1 |- ( A = B -> ( C \ A ) = ( C \ B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1373   e. wcel 2178   {crab 2490   \ cdif 3171

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 615  ax-in2 616  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-ral 2491  df-rab 2495  df-dif 3176

This theorem is referenced by:  difeq12  3294  difeq2i  3296  difeq2d  3299  disjdif2  3547  ssdifeq0  3551  2oconcl  6548  diffitest  7010  diffifi  7017  undifdc  7047  sbthlem2  7086  isbth  7095  difinfinf  7229  ismkvnex  7283  iscld  14690