Theorem alsraln0m 17062

Description: The general "all some" quantifier with class membership as its antecedent holds if and only if ph holds for every x in A and A is inhabited. This is the intuitionistic form of what set.mm states using A =/= (/); see the section comment. (Contributed by Peter Mazsa, 28-Nov-2018.) (Revised by David A. Wheeler, 20-Jul-2026.)

Assertion

Ref	Expression
alsraln0m	|- ( A.E. x ( x e. A -> ph ) <-> ( A. x e. A ph /\ E. x x e. A ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem alsraln0m

Step	Hyp	Ref	Expression
1		df-als 17036	. 2 |- ( A.E. x ( x e. A -> ph ) <-> ( A. x ( x e. A -> ph ) /\ E. x x e. A ) )
2		df-ral 2533	. . . 4 |- ( A. x e. A ph <-> A. x ( x e. A -> ph ) )
3	2	bicomi 132	. . 3 |- ( A. x ( x e. A -> ph ) <-> A. x e. A ph )
4	3	anbi1i 462	. 2 |- ( ( A. x ( x e. A -> ph ) /\ E. x x e. A ) <-> ( A. x e. A ph /\ E. x x e. A ) )
5	1, 4	bitri 184	1 |- ( A.E. x ( x e. A -> ph ) <-> ( A. x e. A ph /\ E. x x e. A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1400   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:  n0alsm  17065  2alsraln0m  17066