Theorem 19.29

Description: Theorem 19.29 of Margaris p. 90. See also 19.29r . (Contributed by NM, 21-Jun-1993) (Proof shortened by Andrew Salmon, 13-May-2011)

	Ref	Expression
Assertion	19.29	⊢ ( ( ∀ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		pm3.2	⊢ ( 𝜑 → ( 𝜓 → ( 𝜑 ∧ 𝜓 ) ) )
2	1	aleximi	⊢ ( ∀ 𝑥 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
3	2	imp	⊢ ( ( ∀ 𝑥 𝜑 ∧ ∃ 𝑥 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )