Users' Mathboxes Mathbox for David A. Wheeler < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  alsi-no-surprise Structured version   Visualization version   GIF version

Theorem alsi-no-surprise 45937
Description: Demonstrate that there is never a "surprise" when using the allsome quantifier, that is, it is never possible for the consequent to be both always true and always false. This uses the definition of df-alsi 45929; the proof itself builds on alimp-no-surprise 45922. For a contrast, see alimp-surprise 45921. (Contributed by David A. Wheeler, 27-Oct-2018.)
Assertion
Ref Expression
alsi-no-surprise ¬ (∀!𝑥(𝜑𝜓) ∧ ∀!𝑥(𝜑 → ¬ 𝜓))

Proof of Theorem alsi-no-surprise
StepHypRef Expression
1 alimp-no-surprise 45922 . 2 ¬ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑)
2 df-alsi 45929 . . . 4 (∀!𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∃𝑥𝜑))
3 df-alsi 45929 . . . 4 (∀!𝑥(𝜑 → ¬ 𝜓) ↔ (∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑))
42, 3anbi12i 630 . . 3 ((∀!𝑥(𝜑𝜓) ∧ ∀!𝑥(𝜑 → ¬ 𝜓)) ↔ ((∀𝑥(𝜑𝜓) ∧ ∃𝑥𝜑) ∧ (∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑)))
5 anandi3r 1104 . . 3 ((∀𝑥(𝜑𝜓) ∧ ∃𝑥𝜑 ∧ ∀𝑥(𝜑 → ¬ 𝜓)) ↔ ((∀𝑥(𝜑𝜓) ∧ ∃𝑥𝜑) ∧ (∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑)))
6 3ancomb 1100 . . 3 ((∀𝑥(𝜑𝜓) ∧ ∃𝑥𝜑 ∧ ∀𝑥(𝜑 → ¬ 𝜓)) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑))
74, 5, 63bitr2i 302 . 2 ((∀!𝑥(𝜑𝜓) ∧ ∀!𝑥(𝜑 → ¬ 𝜓)) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜑 → ¬ 𝜓) ∧ ∃𝑥𝜑))
81, 7mtbir 326 1 ¬ (∀!𝑥(𝜑𝜓) ∧ ∀!𝑥(𝜑 → ¬ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1088  wal 1540  wex 1786  ∀!walsi 45927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1090  df-ex 1787  df-alsi 45929
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator