Theorem difeqri 2160

Description: Inference from membership to difference.

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		df-dif 2049	. 2 |- (A \ B) = {x | (x e. A /\ -. x e. B)}
2		difeqri.1	. . . 4 |- ((x e. A /\ -. x e. B) <-> x e. C)
3	2	bicomi 172	. . 3 |- (x e. C <-> (x e. A /\ -. x e. B))
4	3	abbi2i 1574	. 2 |- C = {x | (x e. A /\ -. x e. B)}
5	1, 4	eqtr4 1498	1 |- (A \ B) = C

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463   \ cdif 2044

This theorem is referenced by:  difdif 2166  ddif 2169  difab 2269

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-dif 2049

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