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Metamath Proof Explorer

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