Theorem elabd3

Description: Membership in a class abstraction, using implicit substitution. Deduction version of elab . (Contributed by Gino Giotto, 12-Oct-2024)

	Ref	Expression
Hypotheses	elabd3.ex	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	elabd3.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	elabd3	⊢ ( 𝜑 → ( 𝐴 ∈ { 𝑥 ∣ 𝜓 } ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		elabd3.ex	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		elabd3.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3		eqidd	⊢ ( 𝜑 → { 𝑥 ∣ 𝜓 } = { 𝑥 ∣ 𝜓 } )
4	1 3 2	elabd2	⊢ ( 𝜑 → ( 𝐴 ∈ { 𝑥 ∣ 𝜓 } ↔ 𝜒 ) )