Theorem difeq2 3234

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 2230	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	notbid 657	. . 3 |- ( A = B -> ( -. x e. A <-> -. x e. B ) )
3	2	rabbidv 2715	. 2 |- ( A = B -> { x e. C | -. x e. A } = { x e. C | -. x e. B } )
4		dfdif2 3124	. 2 |- ( C \ A ) = { x e. C | -. x e. A }
5		dfdif2 3124	. 2 |- ( C \ B ) = { x e. C | -. x e. B }
6	3, 4, 5	3eqtr4g 2224	1 |- ( A = B -> ( C \ A ) = ( C \ B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1343   e. wcel 2136   {crab 2448   \ cdif 3113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2449  df-rab 2453  df-dif 3118

This theorem is referenced by:  difeq12  3235  difeq2i  3237  difeq2d  3240  disjdif2  3487  ssdifeq0  3491  2oconcl  6407  diffitest  6853  diffifi  6860  undifdc  6889  sbthlem2  6923  isbth  6932  difinfinf  7066  ismkvnex  7119  iscld  12743