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	⊢ ¬ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ∧ ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → ¬ 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		als-no-surprise	⊢ ¬ ( ∀∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ∧ ∀∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ¬ 𝜓 ) )
2		dfrals2	⊢ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) )
3		dfrals2	⊢ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → ¬ 𝜓 ) ↔ ∀∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ¬ 𝜓 ) )
4	2 3	anbi12i	⊢ ( ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ∧ ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → ¬ 𝜓 ) ) ↔ ( ∀∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → 𝜓 ) ∧ ∀∃ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ¬ 𝜓 ) ) )
5	1 4	mtbir	⊢ ¬ ( ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ∧ ∀∃ 𝑥 ∈ 𝐴 ( 𝜑 → ¬ 𝜓 ) )