Theorem rals-no-surprise

Description: Demonstrate that there is never a "surprise" when using the allsome quantifier restricted to a class, that is, it is never possible for the consequent to be both always true and always false of the members of A that satisfy the antecedent. This is the restricted counterpart of als-no-surprise , and follows from it by dfrals2 . Note that this holds without any assumption that A is nonempty; that is the point of allsome, since the corresponding claim for the ordinary restricted "for all" fails, as shown in empty-surprise2 . (Contributed by David A. Wheeler, 12-Jul-2026)

	Ref	Expression
Assertion	rals-no-surprise	|- -. ( AE x e. A ( ph -> ps ) /\ AE x e. A ( ph -> -. ps ) )

Proof

Step	Hyp	Ref	Expression
1		als-no-surprise	|- -. ( AE x ( ( x e. A /\ ph ) -> ps ) /\ AE x ( ( x e. A /\ ph ) -> -. ps ) )
2		dfrals2	|- ( AE x e. A ( ph -> ps ) <-> AE x ( ( x e. A /\ ph ) -> ps ) )
3		dfrals2	|- ( AE x e. A ( ph -> -. ps ) <-> AE x ( ( x e. A /\ ph ) -> -. ps ) )
4	2 3	anbi12i	|- ( ( AE x e. A ( ph -> ps ) /\ AE x e. A ( ph -> -. ps ) ) <-> ( AE x ( ( x e. A /\ ph ) -> ps ) /\ AE x ( ( x e. A /\ ph ) -> -. ps ) ) )
5	1 4	mtbir	|- -. ( AE x e. A ( ph -> ps ) /\ AE x e. A ( ph -> -. ps ) )