Theorem difeqri 3248

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 3131	. . 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 183	. 2 |- ( x e. ( A \ B ) <-> x e. C )
4	3	eqriv 2168	1 |- ( A \ B ) = C

Syntax hints:   -. wn 3   /\ wa 103   <-> wb 104   = wceq 1349   e. wcel 2142   \ cdif 3119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153

This theorem depends on definitions:  df-bi 116  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-v 2733  df-dif 3124

This theorem is referenced by:  difdif  3253  ddifnel  3259  difab  3397