Theorem difeq2 3335

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 2298	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	notbid 673	. . 3 |- ( A = B -> ( -. x e. A <-> -. x e. B ) )
3	2	rabbidv 2804	. 2 |- ( A = B -> { x e. C | -. x e. A } = { x e. C | -. x e. B } )
4		dfdif2 3222	. 2 |- ( C \ A ) = { x e. C | -. x e. A }
5		dfdif2 3222	. 2 |- ( C \ B ) = { x e. C | -. x e. B }
6	3, 4, 5	3eqtr4g 2292	1 |- ( A = B -> ( C \ A ) = ( C \ B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   e. wcel 2205   {crab 2526   \ cdif 3211

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 619  ax-in2 620  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-ral 2527  df-rab 2531  df-dif 3216

This theorem is referenced by:  difeq12  3336  difeq2i  3338  difeq2d  3341  disjdif2  3592  ssdifeq0  3596  2oconcl  6685  diffitest  7157  diffifi  7164  undifdc  7197  sbthlem2  7241  isbth  7250  difinfinf  7405  ismkvnex  7459  ballotfilemfval  13173  ballotfilemgval  13211  iscld  15094