Theorem clel4OLD

Description: Obsolete version of clel4 as of 1-Sep-2024. (Contributed by NM, 18-Aug-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypothesis	clel4.1	⊢ 𝐵 ∈ V
Assertion	clel4OLD	⊢ ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		clel4.1	⊢ 𝐵 ∈ V
2		eleq2	⊢ ( 𝑥 = 𝐵 → ( 𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝐵 ) )
3	1 2	ceqsalv	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝑥 ) ↔ 𝐴 ∈ 𝐵 )
4	3	bicomi	⊢ ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐵 → 𝐴 ∈ 𝑥 ) )