Theorem cbvdisjdavw2

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

	Ref	Expression
Hypotheses	cbvdisjdavw2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐶 = 𝐷 )
	cbvdisjdavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
Assertion	cbvdisjdavw2	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvdisjdavw2.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐶 = 𝐷 )
2		cbvdisjdavw2.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐴 = 𝐵 )
3	1	eleq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑡 ∈ 𝐶 ↔ 𝑡 ∈ 𝐷 ) )
4	3 2	cbvrmodavw2	⊢ ( 𝜑 → ( ∃* 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃* 𝑦 ∈ 𝐵 𝑡 ∈ 𝐷 ) )
5	4	albidv	⊢ ( 𝜑 → ( ∀ 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	3bitr4g	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑦 ∈ 𝐵 𝐷 ) )