Theorem rals-no-surprise 50465

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 50464, and follows from it by dfrals2 50448. 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 50441. (Contributed by David A. Wheeler, 12-Jul-2026.)

Assertion

Ref	Expression
rals-no-surprise	⊢ ¬ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓))

Proof of Theorem rals-no-surprise

Step	Hyp	Ref	Expression
1		als-no-surprise 50464	. 2 ⊢ ¬ (∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ∧ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → ¬ 𝜓))
2		dfrals2 50448	. . 3 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
3		dfrals2 50448	. . 3 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → ¬ 𝜓))
4	2, 3	anbi12i 639	. 2 ⊢ ((∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓)) ↔ (∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ∧ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → ¬ 𝜓)))
5	1, 4	mtbir 326	1 ⊢ ¬ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   ∧ wa 400   ∈ wcel 2149  ∀∃wals 50444  ∀∃wrals 50445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-ex 1807  df-ral 3086  df-rex 3096  df-als 50446  df-rals 50447

This theorem is referenced by: (None)