Theorem clel2 1374

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

Hypothesis

Ref	Expression
clel2.1	⊢ A ∈ V

Assertion

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

Distinct variable group(s):   x,A   x,B

Proof of Theorem clel2

Step	Hyp	Ref	Expression
1		clel2.1	. . 3 ⊢ A ∈ V
2		eleq1 1149	. . 3 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
3	1, 2	ceqsalv 1364	. 2 ⊢ (∀x(x = A → x ∈ B) ↔ A ∈ B)
4	3	bicomi 150	1 ⊢ (A ∈ B ↔ ∀x(x = A → x ∈ B))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  snss 1849

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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