Theorem difeq1 3151

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 2647	. 2 |- ( A = B -> { x e. A | -. x e. C } = { x e. B | -. x e. C } )
2		dfdif2 3043	. 2 |- ( A \ C ) = { x e. A | -. x e. C }
3		dfdif2 3043	. 2 |- ( B \ C ) = { x e. B | -. x e. C }
4	1, 2, 3	3eqtr4g 2170	1 |- ( A = B -> ( A \ C ) = ( B \ C ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1312   e. wcel 1461   {crab 2392   \ cdif 3032

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rab 2397  df-dif 3037

This theorem is referenced by:  difeq12  3153  difeq1i  3154  difeq1d  3157  uneqdifeqim  3412  diffitest  6731