Theorem clel4 1897

Description: An alternate definition of class membership when the class is a set.

Hypothesis

Ref	Expression
clel4.1	|- B e. V

Assertion

Ref	Expression
clel4	|- (A e. B <-> A.x(x = B -> A e. x))

Distinct variable groups:   x,A   x,B

Proof of Theorem clel4

Step	Hyp	Ref	Expression
1		clel4.1	. . 3 |- B e. V
2		eleq2 1538	. . 3 |- (x = B -> (A e. x <-> A e. B))
3	1, 2	ceqsalv 1830	. 2 |- (A.x(x = B -> A e. x) <-> A e. B)
4	3	bicomi 172	1 |- (A e. B <-> A.x(x = B -> A e. x))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 956   = wceq 958   e. wcel 960  Vcvv 1814

This theorem is referenced by:  intpr 2567  dfiin2 2592

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  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-v 1815

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