Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  reurex GIF version

Theorem reurex 2642
 Description: Restricted unique existence implies restricted existence. (Contributed by NM, 19-Aug-1999.)
Assertion
Ref Expression
reurex (∃!𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑)

Proof of Theorem reurex
StepHypRef Expression
1 reu5 2641 . 2 (∃!𝑥𝐴 𝜑 ↔ (∃𝑥𝐴 𝜑 ∧ ∃*𝑥𝐴 𝜑))
21simplbi 272 1 (∃!𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∃wrex 2415  ∃!wreu 2416  ∃*wrmo 2417 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-rex 2420  df-reu 2421  df-rmo 2422 This theorem is referenced by:  reu3  2869  prsrriota  7589  elrealeu  7630  ivthinc  12779
 Copyright terms: Public domain W3C validator