Theorem difeqri 3293

Description: Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Hypothesis

Ref	Expression
difeqri.1	|- ( ( x e. A /\ -. x e. B ) <-> x e. C )

Assertion

Ref	Expression
difeqri	|- ( A \ B ) = C

Distinct variable groups:   x, A   x, B   x, C

Proof of Theorem difeqri

Step	Hyp	Ref	Expression
1		eldif 3175	. . 3 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
2		difeqri.1	. . 3 |- ( ( x e. A /\ -. x e. B ) <-> x e. C )
3	1, 2	bitri 184	. 2 |- ( x e. ( A \ B ) <-> x e. C )
4	3	eqriv 2202	1 |- ( A \ B ) = C

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   \ cdif 3163

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-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168

This theorem is referenced by:  difdif  3298  ddifnel  3304  difab  3442