Theorem r19.29r

Description: Restricted quantifier version of 19.29r ; variation of r19.29 . (Contributed by NM, 31-Aug-1999) (Proof shortened by Wolf Lammen, 29-Jun-2023)

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

Proof

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