Theorem r19.29

Description: Restricted quantifier version of 19.29 . See also r19.29r . (Contributed by NM, 31-Aug-1999) (Proof shortened by Andrew Salmon, 30-May-2011) (Proof shortened by Wolf Lammen, 22-Dec-2024)

	Ref	Expression
Assertion	r19.29	⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		ibar	⊢ ( 𝜑 → ( 𝜓 ↔ ( 𝜑 ∧ 𝜓 ) ) )
2	1	ralrexbid	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 ↔ ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) ) )
3	2	biimpa	⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝜓 ) → ∃ 𝑥 ∈ 𝐴 ( 𝜑 ∧ 𝜓 ) )