Theorem difeq1 3244

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 2727	. 2 |- ( A = B -> { x e. A | -. x e. C } = { x e. B | -. x e. C } )
2		dfdif2 3135	. 2 |- ( A \ C ) = { x e. A | -. x e. C }
3		dfdif2 3135	. 2 |- ( B \ C ) = { x e. B | -. x e. C }
4	1, 2, 3	3eqtr4g 2233	1 |- ( A = B -> ( A \ C ) = ( B \ C ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1353   e. wcel 2146   {crab 2457   \ cdif 3124

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rab 2462  df-dif 3129

This theorem is referenced by:  difeq12  3246  difeq1i  3247  difeq1d  3250  uneqdifeqim  3506  diffitest  6877