MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eu3v Structured version   Visualization version   GIF version

Theorem eu3v 2610
Description: An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Add a disjoint variable condition on 𝜑, 𝑦. (Revised by Wolf Lammen, 29-May-2019.)
Assertion
Ref Expression
eu3v (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem eu3v
StepHypRef Expression
1 df-eu 2609 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2 df-mo 2591 . . 3 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
32anbi2i 617 . 2 ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
41, 3bitri 267 1 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wal 1651  wex 1875  ∃*wmo 2589  ∃!weu 2608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 199  df-an 386  df-mo 2591  df-eu 2609
This theorem is referenced by:  eu6  2613  eqeu  3571  reu3  3592  eunexOLD  5060
  Copyright terms: Public domain W3C validator