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	⊢ ¬ ( ∀∃! 𝑥 ( 𝜑 → 𝜓 ) ∧ ∀∃! 𝑥 ( 𝜑 → ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		als-no-surprise	⊢ ¬ ( ∀∃ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∀∃ 𝑥 ( 𝜑 → ¬ 𝜓 ) )
2		alseuals	⊢ ( ∀∃! 𝑥 ( 𝜑 → 𝜓 ) → ∀∃ 𝑥 ( 𝜑 → 𝜓 ) )
3		alseuals	⊢ ( ∀∃! 𝑥 ( 𝜑 → ¬ 𝜓 ) → ∀∃ 𝑥 ( 𝜑 → ¬ 𝜓 ) )
4	2 3	anim12i	⊢ ( ( ∀∃! 𝑥 ( 𝜑 → 𝜓 ) ∧ ∀∃! 𝑥 ( 𝜑 → ¬ 𝜓 ) ) → ( ∀∃ 𝑥 ( 𝜑 → 𝜓 ) ∧ ∀∃ 𝑥 ( 𝜑 → ¬ 𝜓 ) ) )
5	1 4	mto	⊢ ¬ ( ∀∃! 𝑥 ( 𝜑 → 𝜓 ) ∧ ∀∃! 𝑥 ( 𝜑 → ¬ 𝜓 ) )