Theorem n0alsm 17065

Description: If A is inhabited, 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, 20-Jul-2026.)

Assertion

Ref	Expression
n0alsm	|- ( E. x x e. A -> ( A.E. x ( x e. A -> ph ) <-> A. x e. A ph ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem n0alsm

Step	Hyp	Ref	Expression
1		alsraln0m 17062	. 2 |- ( A.E. x ( x e. A -> ph ) <-> ( A. x e. A ph /\ E. x x e. A ) )
2	1	rbaib 933	1 |- ( E. x x e. A -> ( A.E. x ( x e. A -> ph ) <-> A. x e. A ph ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1545   e. wcel 2209   A.wral 2528   A.E.wals 17034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-ral 2533  df-als 17036

This theorem is referenced by: (None)