Theorem clel2 2976

Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
clel2.1	⊢ A ∈ V

Assertion

Ref	Expression
clel2	⊢ (A ∈ B ↔ ∀x(x = A → x ∈ B))

Distinct variable groups:   x,A   x,B

Proof of Theorem clel2

Step	Hyp	Ref	Expression
1		clel2.1	. . 3 ⊢ A ∈ V
2		eleq1 2413	. . 3 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
3	1, 2	ceqsalv 2886	. 2 ⊢ (∀x(x = A → x ∈ B) ↔ A ∈ B)
4	3	bicomi 193	1 ⊢ (A ∈ B ↔ ∀x(x = A → x ∈ B))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2860

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-11 1746  ax-ext 2334

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

This theorem is referenced by:  snss  3839