Theorem bj-clel3gALT

Description: Alternate proof of clel3g . (Contributed by BJ, 1-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	bj-clel3gALT	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elisset	⊢ ( 𝐵 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐵 )
2	1	biantrurd	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ ( ∃ 𝑥 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵 ) ) )
3		19.41v	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵 ) ↔ ( ∃ 𝑥 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵 ) )
4	2 3	bitr4di	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵 ) ) )
5		eleq2	⊢ ( 𝑥 = 𝐵 → ( 𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵 ) )
6	5	bicomd	⊢ ( 𝑥 = 𝐵 → ( 𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝑥 ) )
7	6	pm5.32i	⊢ ( ( 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵 ) ↔ ( 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥 ) )
8	7	exbii	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝐵 ) ↔ ∃ 𝑥 ( 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥 ) )
9	4 8	bitrdi	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐵 ∧ 𝐴 ∈ 𝑥 ) ) )