Theorem cbvdisjdavw

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

	Ref	Expression
Hypothesis	cbvdisjdavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐵 = 𝐶 )
Assertion	cbvdisjdavw	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		cbvdisjdavw.1	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → 𝐵 = 𝐶 )
2	1	eleq2d	⊢ ( ( 𝜑 ∧ 𝑥 = 𝑦 ) → ( 𝑡 ∈ 𝐵 ↔ 𝑡 ∈ 𝐶 ) )
3	2	cbvrmodavw	⊢ ( 𝜑 → ( ∃* 𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ↔ ∃* 𝑦 ∈ 𝐴 𝑡 ∈ 𝐶 ) )
4	3	albidv	⊢ ( 𝜑 → ( ∀ 𝑡 ∃* 𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 ↔ ∀ 𝑡 ∃* 𝑦 ∈ 𝐴 𝑡 ∈ 𝐶 ) )
5		df-disj	⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑡 ∃* 𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 )
6		df-disj	⊢ ( Disj 𝑦 ∈ 𝐴 𝐶 ↔ ∀ 𝑡 ∃* 𝑦 ∈ 𝐴 𝑡 ∈ 𝐶 )
7	4 5 6	3bitr4g	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐴 𝐶 ) )