Metamath Proof Explorer

Theorem 19.31

Description: Theorem 19.31 of Margaris p. 90. See 19.31v for a version requiring fewer axioms. (Contributed by NM, 14-May-1993)

Ref Expression
Hypothesis 19.31.1 ⊢ Ⅎ x ψ
Assertion 19.31 ⊢ ∀ x φ ∨ ψ ↔ ∀ x φ ∨ ψ

Proof

Step Hyp Ref Expression
1 19.31.1 ⊢ Ⅎ x ψ
2 1 19.32 ⊢ ∀ x ψ ∨ φ ↔ ψ ∨ ∀ x φ
3 orcom ⊢ φ ∨ ψ ↔ ψ ∨ φ
4 3 albii ⊢ ∀ x φ ∨ ψ ↔ ∀ x ψ ∨ φ
5 orcom ⊢ ∀ x φ ∨ ψ ↔ ψ ∨ ∀ x φ
6 2 4 5 3bitr4i ⊢ ∀ x φ ∨ ψ ↔ ∀ x φ ∨ ψ