MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mobi Structured version   Visualization version   GIF version

Theorem mobi 2623
Description: Equivalence theorem for the at-most-one quantifier. (Contributed by BJ, 7-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.)
Assertion
Ref Expression
mobi (∀𝑥(𝜑𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))

Proof of Theorem mobi
StepHypRef Expression
1 albiim 1881 . 2 (∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
2 moim 2619 . . . 4 (∀𝑥(𝜓𝜑) → (∃*𝑥𝜑 → ∃*𝑥𝜓))
3 moim 2619 . . . 4 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))
42, 3impbid21d 212 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥(𝜓𝜑) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓)))
54imp 407 . 2 ((∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))
61, 5sylbi 218 1 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526  ∃*wmo 2613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-mo 2615
This theorem is referenced by:  mobiiOLD  2625  mobidv  2626  mobid  2627  eubi  2662
  Copyright terms: Public domain W3C validator