Theorem rals-no-surprise 17056

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 17055, and follows from it by dfrals2 17038. Note that this holds without any assumption that 𝐴 is inhabited; that is the point of allsome, since the corresponding claim for the ordinary restricted "for all" fails when nothing in 𝐴 satisfies 𝜑. (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 17055	. 2 ⊢ ¬ (∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ∧ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → ¬ 𝜓))
2		dfrals2 17038	. . 3 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓))
3		dfrals2 17038	. . 3 ⊢ (∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓) ↔ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → ¬ 𝜓))
4	2, 3	anbi12i 464	. 2 ⊢ ((∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓)) ↔ (∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓) ∧ ∀∃𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → ¬ 𝜓)))
5	1, 4	mtbir 682	1 ⊢ ¬ (∀∃𝑥 ∈ 𝐴(𝜑 → 𝜓) ∧ ∀∃𝑥 ∈ 𝐴(𝜑 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   ∧ wa 104   ∈ wcel 2209  ∀∃wals 17034  ∀∃wrals 17035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie2 1547

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-ral 2533  df-rex 2534  df-als 17036  df-rals 17037

This theorem is referenced by: (None)