Theorem disjeq1d

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

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

Proof

Step	Hyp	Ref	Expression
1		disjeq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		disjeq1	⊢ ( 𝐴 = 𝐵 → ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶 ) )
3	1 2	syl	⊢ ( 𝜑 → ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ Disj 𝑥 ∈ 𝐵 𝐶 ) )