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Metamath Proof Explorer

Theorem 19.30

Description: Theorem 19.30 of Margaris p. 90. (Contributed by NM, 12-Mar-1993) (Proof shortened by Andrew Salmon, 25-May-2011)

Ref Expression
Assertion 19.30 ⊢ ∀ x φ ∨ ψ → ∀ x φ ∨ ∃ x ψ

Proof

Step Hyp Ref Expression
1 exnal ⊢ ∃ x ¬ φ ↔ ¬ ∀ x φ
2 pm2.53 ⊢ φ ∨ ψ → ¬ φ → ψ
3 2 aleximi ⊢ ∀ x φ ∨ ψ → ∃ x ¬ φ → ∃ x ψ
4 1 3 biimtrrid ⊢ ∀ x φ ∨ ψ → ¬ ∀ x φ → ∃ x ψ
5 4 orrd ⊢ ∀ x φ ∨ ψ → ∀ x φ ∨ ∃ x ψ