Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mopickr Structured version   Visualization version   GIF version

Theorem mopickr 37536
Description: "At most one" picks a variable value, eliminating an existential quantifier. The proof begins with references *2.21 (pm2.21 123) and *14.26 (eupickbi 2631) from [WhiteheadRussell] p. 104 and p. 183. (Contributed by Peter Mazsa, 18-Nov-2024.) (Proof modification is discouraged.)
Assertion
Ref Expression
mopickr ((∃*𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜓𝜑))

Proof of Theorem mopickr
StepHypRef Expression
1 exancom 1863 . . 3 (∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
2 moeu2 37535 . . . 4 (∃*𝑥𝜓 ↔ (¬ ∃𝑥𝜓 ∨ ∃!𝑥𝜓))
3 19.8a 2173 . . . . . . . 8 (𝜓 → ∃𝑥𝜓)
43con3i 154 . . . . . . 7 (¬ ∃𝑥𝜓 → ¬ 𝜓)
5 pm2.21 123 . . . . . . 7 𝜓 → (𝜓𝜑))
64, 5syl 17 . . . . . 6 (¬ ∃𝑥𝜓 → (𝜓𝜑))
76a1d 25 . . . . 5 (¬ ∃𝑥𝜓 → (∃𝑥(𝜓𝜑) → (𝜓𝜑)))
8 eupickbi 2631 . . . . . 6 (∃!𝑥𝜓 → (∃𝑥(𝜓𝜑) ↔ ∀𝑥(𝜓𝜑)))
9 sp 2175 . . . . . 6 (∀𝑥(𝜓𝜑) → (𝜓𝜑))
108, 9syl6bi 253 . . . . 5 (∃!𝑥𝜓 → (∃𝑥(𝜓𝜑) → (𝜓𝜑)))
117, 10jaoi 854 . . . 4 ((¬ ∃𝑥𝜓 ∨ ∃!𝑥𝜓) → (∃𝑥(𝜓𝜑) → (𝜓𝜑)))
122, 11sylbi 216 . . 3 (∃*𝑥𝜓 → (∃𝑥(𝜓𝜑) → (𝜓𝜑)))
131, 12biimtrid 241 . 2 (∃*𝑥𝜓 → (∃𝑥(𝜑𝜓) → (𝜓𝜑)))
1413imp 406 1 ((∃*𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 844  wal 1538  wex 1780  ∃*wmo 2531  ∃!weu 2561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-11 2153  ax-12 2170
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-mo 2533  df-eu 2562
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator