Theorem difeqri2 14341

Description: Inference from membership to difference. (The proof was shortened by Andrew Salmon, 9-Jul-2011.)

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		eldif 2609	. . . 4 |- (x e. (A \ B) <-> (x e. A /\ -. x e. B))
2	1	bibi1i 671	. . 3 |- ((x e. (A \ B) <-> x e. C) <-> ((x e. A /\ -. x e. B) <-> x e. C))
3	2	albii 1346	. 2 |- (A.x(x e. (A \ B) <-> x e. C) <-> A.x((x e. A /\ -. x e. B) <-> x e. C))
4		dfcleq 1878	. . 3 |- ((A \ B) = C <-> A.x(x e. (A \ B) <-> x e. C))
5	4	biimpri 169	. 2 |- (A.x(x e. (A \ B) <-> x e. C) -> (A \ B) = C)
6	3, 5	sylbir 218	1 |- (A.x((x e. A /\ -. x e. B) <-> x e. C) -> (A \ B) = C)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300   \ cdif 2590

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-dif 2597

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