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 =/= (/) -> ( AE x ( x e. A -> ph ) <-> A. x e. A ph ) )

Proof

Step	Hyp	Ref	Expression
1		alsraln0	|- ( AE x ( x e. A -> ph ) <-> ( A. x e. A ph /\ A =/= (/) ) )
2	1	rbaib	|- ( A =/= (/) -> ( AE x ( x e. A -> ph ) <-> A. x e. A ph ) )