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

Theorem rexsng 4363
Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
Hypothesis
Ref Expression
ralsng.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rexsng (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rexsng
StepHypRef Expression
1 rexsns 4361 . 2 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
2 ralsng.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32sbcieg 3609 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
41, 3syl5bb 272 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wcel 2139  wrex 3051  [wsbc 3576  {csn 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-v 3342  df-sbc 3577  df-sn 4322
This theorem is referenced by:  rexsn  4367  rexprg  4379  rextpg  4381  iunxsng  4754  frirr  5243  frsn  5346  imasng  5645  scshwfzeqfzo  13772  dvdsprmpweqnn  15791  mnd1  17532  grp1  17723  1loopgrvd0  26610  1egrvtxdg0  26617  nfrgr2v  27426  1vwmgr  27430  ballotlemfc0  30863  ballotlemfcc  30864  bj-restsn  33341  elpaddat  35593  elpadd2at  35595  brfvidRP  38482  iccelpart  41879  zlidlring  42438  lco0  42726
  Copyright terms: Public domain W3C validator