Theorem difeqri2 8703

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 518	. . . . 5 |- (((x e. A /\ -. x e. B) <-> x e. C) <-> (x e. C <-> (x e. A /\ -. x e. B)))
2	1	albii 975	. . . 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 1544	. . 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 2020	. 2 |- (A \ B) = {x | (x e. A /\ -. x e. B)}
7	5, 6	syl6reqr 1502	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 950   = wceq 1099   e. wcel 1105  {cab 1440   \ cdif 2015

This theorem is referenced by:  cdrci 8738

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-dif 2020

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