Theorem ralsex

Description: The consequent of an "all some" restricted to a class is witnessed: some member of A satisfying ph also satisfies ps . Restricted counterpart of alsex . (Contributed by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Assertion	ralsex	⊢ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ∃ 𝑥 ∈ 𝐴 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		df-rals	⊢ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 ∈ 𝐴 𝜑 ) )
2		rexim	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∃ 𝑥 ∈ 𝐴 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 ) )
3	2	imp	⊢ ( ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ∧ ∃ 𝑥 ∈ 𝐴 𝜑 ) → ∃ 𝑥 ∈ 𝐴 𝜓 )
4	1 3	sylbi	⊢ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ∃ 𝑥 ∈ 𝐴 𝜓 )