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Metamath Proof Explorer

Theorem 19.32v

Description: Version of 19.32 with a disjoint variable condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020)

Ref Expression
Assertion 19.32v ⊢ ∀ x φ ∨ ψ ↔ φ ∨ ∀ x ψ

Proof

Step Hyp Ref Expression
1 19.21v ⊢ ∀ x ¬ φ → ψ ↔ ¬ φ → ∀ x ψ
2 df-or ⊢ φ ∨ ψ ↔ ¬ φ → ψ
3 2 albii ⊢ ∀ x φ ∨ ψ ↔ ∀ x ¬ φ → ψ
4 df-or ⊢ φ ∨ ∀ x ψ ↔ ¬ φ → ∀ x ψ
5 1 3 4 3bitr4i ⊢ ∀ x φ ∨ ψ ↔ φ ∨ ∀ x ψ