Theorem disjeq12d

Description: Equality theorem for disjoint collection. (Contributed by Mario Carneiro, 14-Nov-2016)

	Ref	Expression
Hypotheses	disjeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	disjeq12d.1	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	disjeq12d	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		disjeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		disjeq12d.1	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1	disjeq1d	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶 ) )
4	2	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐵 ) → 𝐶 = 𝐷 )
5	4	disjeq2dv	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐵 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷 ) )
6	3 5	bitrd	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐷 ) )