Theorem disjeq12dv

Description: Equality theorem for disjoint collection. Deduction version. (Contributed by GG, 1-Sep-2025)

	Ref	Expression
Hypotheses	disjeq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	disjeq12dv.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	disjeq12dv	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		disjeq12dv.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		disjeq12dv.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1	eleq2d	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵 ) )
4	3	anbi1d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶 ) ) )
5	4	mobidv	⊢ ( 𝜑 → ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶 ) ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶 ) ) )
6		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐶 ) )
7		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐵 ∧ 𝑡 ∈ 𝐶 ) )
8	5 6 7	3bitr4g	⊢ ( 𝜑 → ( ∃* 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∃* 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 ) )
9	8	albidv	⊢ ( 𝜑 → ( ∀ 𝑡 ∃* 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 ↔ ∀ 𝑡 ∃* 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 ) )
10		df-disj	⊢ ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀ 𝑡 ∃* 𝑥 ∈ 𝐴 𝑡 ∈ 𝐶 )
11		df-disj	⊢ ( Disj 𝑥 ∈ 𝐵 𝐶 ↔ ∀ 𝑡 ∃* 𝑥 ∈ 𝐵 𝑡 ∈ 𝐶 )
12	9 10 11	3bitr4g	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶 ) )
13	2	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 = 𝐷 )
14	13	disjeq2dv	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐵 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷 ) )
15	12 14	bitrd	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷 ) )