Theorem clel2g

Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993) (Revised by BJ, 12-Feb-2022)

	Ref	Expression
Assertion	clel2g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		nfv	⊢ Ⅎ 𝑥 𝐴 ∈ 𝐵
2		eleq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
3	1 2	ceqsalg	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) ↔ 𝐴 ∈ 𝐵 ) )
4	3	bicomd	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) ) )