Theorem difeq2 3341

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 2302	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	notbid 677	. . 3 |- ( A = B -> ( -. x e. A <-> -. x e. B ) )
3	2	rabbidv 2810	. 2 |- ( A = B -> { x e. C | -. x e. A } = { x e. C | -. x e. B } )
4		dfdif2 3228	. 2 |- ( C \ A ) = { x e. C | -. x e. A }
5		dfdif2 3228	. 2 |- ( C \ B ) = { x e. C | -. x e. B }
6	3, 4, 5	3eqtr4g 2296	1 |- ( A = B -> ( C \ A ) = ( C \ B ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   = wceq 1402   e. wcel 2209   {crab 2532   \ cdif 3217

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-rab 2537  df-dif 3222

This theorem is used by:  difeq12  3342  difeq2i  3344  difeq2d  3347  disjdif2  3606  ssdifeq0  3610  2oconcl  6712  diffitest  7191  diffifi  7198  undifdc  7231  sbthlem2  7275  isbth  7284  difinfinf  7441  ismkvnex  7495  ballotfilemfval  13229  ballotfilemgval  13267  iscld  15204  wexmiddiffilem  17043  wexmiddifxy  17046