Theorem cbvrexvw2

Description: Change bound variable and domain in the restricted existential quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvrexvw2.1	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
	cbvrexvw2.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	cbvrexvw2	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		cbvrexvw2.1	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
2		cbvrexvw2.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
4	1	eleq2d	⊢ ( 𝑥 = 𝑦 → ( 𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
5	3 4	bitrd	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵 ) )
6	5 2	anbi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) ) )
7	6	cbvexvw	⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) )
8		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
9		df-rex	⊢ ( ∃ 𝑦 ∈ 𝐵 𝜓 ↔ ∃ 𝑦 ( 𝑦 ∈ 𝐵 ∧ 𝜓 ) )
10	7 8 9	3bitr4i	⊢ ( ∃ 𝑥 ∈ 𝐴 𝜑 ↔ ∃ 𝑦 ∈ 𝐵 𝜓 )