Theorem difeqri 3284

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 3166	. . 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 2193	1 |- ( A \ B ) = C

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   \ cdif 3154

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159

This theorem is referenced by:  difdif  3289  ddifnel  3295  difab  3433