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

Theorem rexsn 4682
Description: Convert an existential quantification restricted to a singleton to a substitution. (Contributed by Jeff Madsen, 5-Jan-2011.)
Hypotheses
Ref Expression
ralsn.1 𝐴 ∈ V
ralsn.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rexsn (∃𝑥 ∈ {𝐴}𝜑𝜓)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexsn
StepHypRef Expression
1 ralsn.1 . 2 𝐴 ∈ V
2 ralsn.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32rexsng 4674 . 2 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑𝜓))
41, 3ax-mp 5 1 (∃𝑥 ∈ {𝐴}𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  wrex 3071  Vcvv 3475  {csn 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-sn 4625
This theorem is referenced by:  elsnres  6016  oarec  8550  snec  8762  zornn0g  10487  fpwwe2lem12  10624  elreal  11113  hashge2el2difr  14429  vdwlem6  16906  pmatcollpw3fi1  22259  restsn  22643  snclseqg  23589  ust0  23693  0slt1s  27297  cuteq1  27301  made0  27335  cofcutr  27378  mulsrid  27536  grplsm0l  32471  esum2dlem  33021  eulerpartlemgh  33308  eldm3  34662  poimirlem28  36421  heiborlem3  36587  tfsconcatrn  41963
  Copyright terms: Public domain W3C validator