Theorem cbvrabv2w

Description: A more general version of cbvrabv . Version of cbvrabv2 with a disjoint variable condition, which does not require ax-13 . (Contributed by Glauco Siliprandi, 23-Oct-2021) (Revised by Gino Giotto, 16-Apr-2024)

	Ref	Expression
Hypotheses	cbvrabv2w.1	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
	cbvrabv2w.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvrabv2w	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑦 ∈ 𝐵 ∣ 𝜓 }

Proof

Step	Hyp	Ref	Expression
1		cbvrabv2w.1	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
2		cbvrabv2w.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3		nfcv	⊢ Ⅎ 𝑦 𝐴
4		nfcv	⊢ Ⅎ 𝑥 𝐵
5		nfv	⊢ Ⅎ 𝑦 𝜑
6		nfv	⊢ Ⅎ 𝑥 𝜓
7	3 4 5 6 1 2	cbvrabcsfw	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑦 ∈ 𝐵 ∣ 𝜓 }