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

Theorem eubi 2582
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 1844 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
2 mobi 2545 . . 3 (∀𝑥(𝜑𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))
31, 2anbi12d 632 . 2 (∀𝑥(𝜑𝜓) → ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜓 ∧ ∃*𝑥𝜓)))
4 df-eu 2567 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
5 df-eu 2567 . 2 (∃!𝑥𝜓 ↔ (∃𝑥𝜓 ∧ ∃*𝑥𝜓))
63, 4, 53bitr4g 314 1 (∀𝑥(𝜑𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wex 1776  ∃*wmo 2536  ∃!weu 2566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-mo 2538  df-eu 2567
This theorem is referenced by:  eubidv  2584  eubid  2585
  Copyright terms: Public domain W3C validator