Theorem difeqri2 10438

Description: Inference from membership to difference.

Assertion

Ref	Expression
difeqri2	|- (A.x((x e. A /\ -. x e. B) <-> x e. C) -> (A \ B) = C)

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

Proof of Theorem difeqri2

Step	Hyp	Ref	Expression
1		bicom 522	. . . . 5 |- (((x e. A /\ -. x e. B) <-> x e. C) <-> (x e. C <-> (x e. A /\ -. x e. B)))
2	1	albii 1001	. . . 4 |- (A.x((x e. A /\ -. x e. B) <-> x e. C) <-> A.x(x e. C <-> (x e. A /\ -. x e. B)))
3	2	biimp 151	. . 3 |- (A.x((x e. A /\ -. x e. B) <-> x e. C) -> A.x(x e. C <-> (x e. A /\ -. x e. B)))
4		abeq2 1571	. . 3 |- (C = {x | (x e. A /\ -. x e. B)} <-> A.x(x e. C <-> (x e. A /\ -. x e. B)))
5	3, 4	sylibr 200	. 2 |- (A.x((x e. A /\ -. x e. B) <-> x e. C) -> C = {x | (x e. A /\ -. x e. B)})
6		df-dif 2052	. 2 |- (A \ B) = {x | (x e. A /\ -. x e. B)}
7	5, 6	syl6reqr 1529	1 |- (A.x((x e. A /\ -. x e. B) <-> x e. C) -> (A \ B) = C)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  {cab 1466   \ cdif 2047

This theorem is referenced by:  cdrci 10480

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-dif 2052

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