Theorem difeq2 3056

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 2101	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	notbid 592	. . 3 |- ( A = B -> ( -. x e. A <-> -. x e. B ) )
3	2	rabbidv 2549	. 2 |- ( A = B -> { x e. C | -. x e. A } = { x e. C | -. x e. B } )
4		dfdif2 2926	. 2 |- ( C \ A ) = { x e. C | -. x e. A }
5		dfdif2 2926	. 2 |- ( C \ B ) = { x e. C | -. x e. B }
6	3, 4, 5	3eqtr4g 2097	1 |- ( A = B -> ( C \ A ) = ( C \ B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1243   e. wcel 1393   {crab 2310   \ cdif 2914

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-rab 2315  df-dif 2920

This theorem is referenced by:  difeq12  3057  difeq2i  3059  difeq2d  3062  ssdifeq0  3305  2oconcl  6022  diffitest  6344  diffifi  6351