Theorem difeq1 3318

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

Assertion

Ref	Expression
difeq1	|- ( A = B -> ( A \ C ) = ( B \ C ) )

Proof of Theorem difeq1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabeq 2794	. 2 |- ( A = B -> { x e. A | -. x e. C } = { x e. B | -. x e. C } )
2		dfdif2 3208	. 2 |- ( A \ C ) = { x e. A | -. x e. C }
3		dfdif2 3208	. 2 |- ( B \ C ) = { x e. B | -. x e. C }
4	1, 2, 3	3eqtr4g 2289	1 |- ( A = B -> ( A \ C ) = ( B \ C ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1397   e. wcel 2202   {crab 2514   \ cdif 3197

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-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-dif 3202

This theorem is referenced by:  difeq12  3320  difeq1i  3321  difeq1d  3324  uneqdifeqim  3580  diffitest  7075  fundm2domnop0  11108