Theorem class2seteq 2725

Description: Equality theorem based on class2set 2724. (The proof was shortened by Raph Levien, 30-Jun-2006.)

Assertion

Ref	Expression
class2seteq	|- (A e. B -> {x e. A | A e. V} = A)

Distinct variable group:   x,A

Proof of Theorem class2seteq

Step	Hyp	Ref	Expression
1		elisset 1808	. 2 |- (A e. B -> A e. V)
2		ax-1 4	. . . . 5 |- (A e. V -> (x e. A -> A e. V))
3	2	r19.21aiv 1705	. . . 4 |- (A e. V -> A.x e. A A e. V)
4		rabid2 1762	. . . 4 |- (A = {x e. A | A e. V} <-> A.x e. A A e. V)
5	3, 4	sylibr 200	. . 3 |- (A e. V -> A = {x e. A | A e. V})
6	5	eqcomd 1472	. 2 |- (A e. V -> {x e. A | A e. V} = A)
7	1, 6	syl 10	1 |- (A e. B -> {x e. A | A e. V} = A)

Syntax hints:   -> wi 3   = wceq 953   e. wcel 955  A.wral 1637  {crab 1640  Vcvv 1802

This theorem is referenced by:  fsum1s 6947  fsump1s 6951

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ral 1641  df-rab 1644  df-v 1803

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