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

Theorem eubi 2603
 Description: Equivalence theorem for the unique existential quantifier. Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) Reduce dependencies on axioms. (Revised by BJ, 7-Oct-2022.)
Assertion
Ref Expression
eubi (∀𝑥(𝜑𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))

Proof of Theorem eubi
StepHypRef Expression
1 exbi 1848 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
2 mobi 2564 . . 3 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))
31, 2anbi12d 633 . 2 (∀𝑥(𝜑𝜓) → ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜓 ∧ ∃*𝑥𝜓)))
4 df-eu 2588 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
5 df-eu 2588 . 2 (∃!𝑥𝜓 ↔ (∃𝑥𝜓 ∧ ∃*𝑥𝜓))
63, 4, 53bitr4g 317 1 (∀𝑥(𝜑𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  ∃*wmo 2555  ∃!weu 2587 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2557  df-eu 2588 This theorem is referenced by:  eubiiOLD  2605  eubidv  2606  eubid  2607
 Copyright terms: Public domain W3C validator