Theorem clel3g 2977

Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)

Assertion

Ref	Expression
clel3g	|- (B e. V -> (A e. B <-> E.x(x = B /\ A e. x)))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   V(x)

Proof of Theorem clel3g

Step	Hyp	Ref	Expression
1		eleq2 2414	. . 3 |- (x = B -> (A e. x <-> A e. B))
2	1	ceqsexgv 2972	. 2 |- (B e. V -> (E.x(x = B /\ A e. x) <-> A e. B))
3	2	bicomd 192	1 |- (B e. V -> (A e. B <-> E.x(x = B /\ A e. x)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  clel3  2978  dfiun2g  4000