Theorem cbvdisjvw2

Description: Change bound variable and domain in a disjoint collection, using implicit substitution. (Contributed by GG, 14-Aug-2025)

	Ref	Expression
Hypotheses	cbvdisjvw2.1	⊢ ( 𝑥 = 𝑦 → 𝐶 = 𝐷 )
	cbvdisjvw2.2	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
Assertion	cbvdisjvw2	⊢ ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		cbvdisjvw2.1	⊢ ( 𝑥 = 𝑦 → 𝐶 = 𝐷 )
2		cbvdisjvw2.2	⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 )
3	1	eleq2d	⊢ ( 𝑥 = 𝑦 → ( 𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷 ) )
4	2 3	cbvrmovw2	⊢ ( ∃* 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃* 𝑦 ∈ 𝐵 𝑡 ∈ 𝐷 )
5	4	albii	⊢ ( ∀ 𝑡 ∃* 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀ 𝑡 ∃* 𝑦 ∈ 𝐵 𝑡 ∈ 𝐷 )
6		df-disj	⊢ ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀ 𝑡 ∃* 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 )
7		df-disj	⊢ ( Disj 𝑦 ∈ 𝐵 𝐷 ↔ ∀ 𝑡 ∃* 𝑦 ∈ 𝐵 𝑡 ∈ 𝐷 )
8	5 6 7	3bitr4i	⊢ ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷 )