Theorem clel2g

Description: Alternate definition of membership when the member is a set. (Contributed by NM, 18-Aug-1993) Strengthen from sethood hypothesis to sethood antecedent (Revised by BJ, 12-Feb-2022) Avoid ax-12 . (Revised by BJ, 1-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		elisset	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
2		biimt	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( 𝐴 ∈ 𝐵 ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵 ) ) )
3	1 2	syl	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵 ) ) )
4		19.23v	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝐴 ∈ 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵 ) )
5		eleq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
6	5	bicomd	⊢ ( 𝑥 = 𝐴 → ( 𝐴 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵 ) )
7	6	pm5.74i	⊢ ( ( 𝑥 = 𝐴 → 𝐴 ∈ 𝐵 ) ↔ ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) )
8	7	albii	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝐴 ∈ 𝐵 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) )
9	4 8	bitr3i	⊢ ( ( ∃ 𝑥 𝑥 = 𝐴 → 𝐴 ∈ 𝐵 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) )
10	3 9	bitrdi	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) ) )