Theorem elabd2

Description: Membership in a class abstraction, using implicit substitution. Deduction version of elab . (Contributed by Gino Giotto, 12-Oct-2024) (Revised by BJ, 16-Oct-2024)

	Ref	Expression
Hypotheses	elabd2.ex	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	elabd2.eq	⊢ ( 𝜑 → 𝐵 = { 𝑥 ∣ 𝜓 } )
	elabd2.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
Assertion	elabd2	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐵 ↔ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		elabd2.ex	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		elabd2.eq	⊢ ( 𝜑 → 𝐵 = { 𝑥 ∣ 𝜓 } )
3		elabd2.is	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
4	2	eleq2d	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐵 ↔ 𝐴 ∈ { 𝑥 ∣ 𝜓 } ) )
5		elab6g	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ { 𝑥 ∣ 𝜓 } ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ) )
6	4 5	sylan9bb	⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∈ 𝐵 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ) )
7		elisset	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
8	3	pm5.74da	⊢ ( 𝜑 → ( ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( 𝑥 = 𝐴 → 𝜒 ) ) )
9	8	albidv	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜒 ) ) )
10		19.23v	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜒 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜒 ) )
11	9 10	bitrdi	⊢ ( 𝜑 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜒 ) ) )
12		pm5.5	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ( ( ∃ 𝑥 𝑥 = 𝐴 → 𝜒 ) ↔ 𝜒 ) )
13	11 12	sylan9bb	⊢ ( ( 𝜑 ∧ ∃ 𝑥 𝑥 = 𝐴 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ 𝜒 ) )
14	7 13	sylan2	⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ 𝜒 ) )
15	6 14	bitrd	⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑉 ) → ( 𝐴 ∈ 𝐵 ↔ 𝜒 ) )
16	1 15	mpdan	⊢ ( 𝜑 → ( 𝐴 ∈ 𝐵 ↔ 𝜒 ) )