Theorem disjeq2

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

	Ref	Expression
Assertion	disjeq2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		eqimss2	⊢ ( 𝐵 = 𝐶 → 𝐶 ⊆ 𝐵 )
2	1	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀ 𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 )
3		disjss2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → ( Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ 𝐴 𝐶 ) )
4	2 3	syl	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ( Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ 𝐴 𝐶 ) )
5		eqimss	⊢ ( 𝐵 = 𝐶 → 𝐵 ⊆ 𝐶 )
6	5	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
7		disjss2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ( Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵 ) )
8	6 7	syl	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ( Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵 ) )
9	4 8	impbid	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐶 → ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑥 ∈ 𝐴 𝐶 ) )