Theorem n0als

Description: If A is not empty, then the general "all some" quantifier with class membership as its antecedent reduces to the assertion that ph holds for every x in A . (Contributed by Peter Mazsa, 19-Dec-2018) (Revised by David A. Wheeler, 15-Jul-2026)

	Ref	Expression
Assertion	n0als	⊢ ( 𝐴 ≠ ∅ → ( ∀∃ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐴 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		alsraln0	⊢ ( ∀∃ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( ∀ 𝑥 ∈ 𝐴 𝜑 ∧ 𝐴 ≠ ∅ ) )
2	1	rbaib	⊢ ( 𝐴 ≠ ∅ → ( ∀∃ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐴 𝜑 ) )