Theorem cbvdisjv

Description: Change bound variables in a disjoint collection. (Contributed by Mario Carneiro, 11-Dec-2016)

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

Proof

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