Metamath Proof Explorer

Theorem ralunsn

Description: Restricted quantification over the union of a set and a singleton, using implicit substitution. (Contributed by Paul Chapman, 17-Nov-2012) (Revised by Mario Carneiro, 23-Apr-2015)

Ref Expression
Hypothesis ralunsn.1 ⊢ x = B → φ ↔ ψ
Assertion ralunsn ⊢ B ∈ C → ∀ x ∈ A ∪ B φ ↔ ∀ x ∈ A φ ∧ ψ

Proof

Step Hyp Ref Expression
1 ralunsn.1 ⊢ x = B → φ ↔ ψ
2 ralunb ⊢ ∀ x ∈ A ∪ B φ ↔ ∀ x ∈ A φ ∧ ∀ x ∈ B φ
3 1 ralsng ⊢ B ∈ C → ∀ x ∈ B φ ↔ ψ
4 3 anbi2d ⊢ B ∈ C → ∀ x ∈ A φ ∧ ∀ x ∈ B φ ↔ ∀ x ∈ A φ ∧ ψ
5 2 4 syl5bb ⊢ B ∈ C → ∀ x ∈ A ∪ B φ ↔ ∀ x ∈ A φ ∧ ψ