Theorem clel2 2671

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 2097	. . 3 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
3	1, 2	ceqsalv 2578	. 2 ⊢ (∀x(x = A → x ∈ B) ↔ A ∈ B)
4	3	bicomi 123	1 ⊢ (A ∈ B ↔ ∀x(x = A → x ∈ B))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1240   = wceq 1242   ∈ wcel 1390  Vcvv 2551

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  snss  3485