Theorem 19.29r

Description: Variation of 19.29 . (Contributed by NM, 18-Aug-1993) (Proof shortened by Wolf Lammen, 12-Nov-2020)

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

Proof

Step	Hyp	Ref	Expression
1		pm3.21	⊢ ( 𝜓 → ( 𝜑 → ( 𝜑 ∧ 𝜓 ) ) )
2	1	aleximi	⊢ ( ∀ 𝑥 𝜓 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) ) )
3	2	impcom	⊢ ( ( ∃ 𝑥 𝜑 ∧ ∀ 𝑥 𝜓 ) → ∃ 𝑥 ( 𝜑 ∧ 𝜓 ) )