Theorem alseu-no-surprise 17153

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 17121 by alseuals 17139. See als-no-surprise 17121 for why ordinary "for all" with implication has no such property. (Contributed by David A. Wheeler, 22-Jul-2026.)

Assertion

Ref	Expression
alseu-no-surprise	⊢ ¬ (∀∃!𝑥(𝜑 → 𝜓) ∧ ∀∃!𝑥(𝜑 → ¬ 𝜓))

Proof of Theorem alseu-no-surprise

Step	Hyp	Ref	Expression
1		als-no-surprise 17121	. 2 ⊢ ¬ (∀∃𝑥(𝜑 → 𝜓) ∧ ∀∃𝑥(𝜑 → ¬ 𝜓))
2		alseuals 17139	. . 3 ⊢ (∀∃!𝑥(𝜑 → 𝜓) → ∀∃𝑥(𝜑 → 𝜓))
3		alseuals 17139	. . 3 ⊢ (∀∃!𝑥(𝜑 → ¬ 𝜓) → ∀∃𝑥(𝜑 → ¬ 𝜓))
4	2, 3	anim12i 338	. 2 ⊢ ((∀∃!𝑥(𝜑 → 𝜓) ∧ ∀∃!𝑥(𝜑 → ¬ 𝜓)) → (∀∃𝑥(𝜑 → 𝜓) ∧ ∀∃𝑥(𝜑 → ¬ 𝜓)))
5	1, 4	mto 672	1 ⊢ ¬ (∀∃!𝑥(𝜑 → 𝜓) ∧ ∀∃!𝑥(𝜑 → ¬ 𝜓))

Syntax hints:  ¬ wn 3   ∧ wa 104  ∀∃wals 17100  ∀∃!walseu 17134

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-als 17102  df-alseu 17136

This theorem is referenced by: (None)