Theorem alseu-no-surprise

Description: Demonstrate that there is never a "surprise" when using the "all some one" quantifier, that is, it is never possible for the consequent to be both always true and always false. This follows from als-no-surprise by alseuals . For a contrast, see alimp-surprise . (Contributed by David A. Wheeler, 21-Jul-2026)

	Ref	Expression
Assertion	alseu-no-surprise	|- -. ( AE! x ( ph -> ps ) /\ AE! x ( ph -> -. ps ) )

Proof

Step	Hyp	Ref	Expression
1		als-no-surprise	|- -. ( AE x ( ph -> ps ) /\ AE x ( ph -> -. ps ) )
2		alseuals	|- ( AE! x ( ph -> ps ) -> AE x ( ph -> ps ) )
3		alseuals	|- ( AE! x ( ph -> -. ps ) -> AE x ( ph -> -. ps ) )
4	2 3	anim12i	|- ( ( AE! x ( ph -> ps ) /\ AE! x ( ph -> -. ps ) ) -> ( AE x ( ph -> ps ) /\ AE x ( ph -> -. ps ) ) )
5	1 4	mto	|- -. ( AE! x ( ph -> ps ) /\ AE! x ( ph -> -. ps ) )