Theorem elrab2w

Description: Membership in a restricted class abstraction. This is to elrab2 what elab2gw is to elab2g . (Contributed by SN, 2-Sep-2024)

	Ref	Expression
Hypotheses	elrab2w.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
	elrab2w.2	⊢ ( 𝑦 = 𝐴 → ( 𝜓 ↔ 𝜒 ) )
	elrab2w.3	⊢ 𝐶 = { 𝑥 ∈ 𝐵 ∣ 𝜑 }
Assertion	elrab2w	⊢ ( 𝐴 ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		elrab2w.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		elrab2w.2	⊢ ( 𝑦 = 𝐴 → ( 𝜓 ↔ 𝜒 ) )
3		elrab2w.3	⊢ 𝐶 = { 𝑥 ∈ 𝐵 ∣ 𝜑 }
4		elex	⊢ ( 𝐴 ∈ 𝐶 → 𝐴 ∈ V )
5		elex	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ∈ V )
6	5	adantr	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜒 ) → 𝐴 ∈ V )
7		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵 ) )
8	7 1	anbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) )
9		eleq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵 ) )
10	9 2	anbi12d	⊢ ( 𝑦 = 𝐴 → ( ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜒 ) ) )
11		df-rab	⊢ { 𝑥 ∈ 𝐵 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) }
12	3 11	eqtri	⊢ 𝐶 = { 𝑥 ∣ ( 𝑥 ∈ 𝐵 ∧ 𝜑 ) }
13	8 10 12	elab2gw	⊢ ( 𝐴 ∈ V → ( 𝐴 ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜒 ) ) )
14	4 6 13	pm5.21nii	⊢ ( 𝐴 ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜒 ) )