Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rexsns Structured version   Unicode version

Theorem rexsns 4035
 Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)
Assertion
Ref Expression
rexsns
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem rexsns
StepHypRef Expression
1 elsn 4016 . . . 4
21anbi1i 699 . . 3
32exbii 1714 . 2
4 df-rex 2788 . 2
5 sbc5 3330 . 2
63, 4, 53bitr4i 280 1
 Colors of variables: wff setvar class Syntax hints:   wb 187   wa 370   wceq 1437  wex 1659   wcel 1870  wrex 2783  wsbc 3305  csn 4002 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-12 1907  ax-13 2055  ax-ext 2407 This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-rex 2788  df-v 3089  df-sbc 3306  df-sn 4003 This theorem is referenced by:  rexsng  4038  r19.12sn  4068  poimirlem25  31669  rexsngf  37032
 Copyright terms: Public domain W3C validator