Theorem clel2 2863

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 2233	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
3	1, 2	ceqsalv 2760	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵) ↔ 𝐴 ∈ 𝐵)
4	3	bicomi 131	1 ⊢ (𝐴 ∈ 𝐵 ↔ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1346   = wceq 1348   ∈ wcel 2141  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  snss  3709