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

Theorem rexex 3001
Description: Restricted existence implies existence. (Contributed by NM, 11-Nov-1995.)
Assertion
Ref Expression
rexex (∃𝑥𝐴 𝜑 → ∃𝑥𝜑)

Proof of Theorem rexex
StepHypRef Expression
1 df-rex 2917 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 exsimpr 1795 . 2 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥𝜑)
31, 2sylbi 207 1 (∃𝑥𝐴 𝜑 → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wex 1703  wcel 1989  wrex 2912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704  df-rex 2917
This theorem is referenced by:  reu3  3394  rmo2i  3525  dffo5  6374  nqerf  9749  supsrlem  9929  vdwmc2  15677  toprntopon  20723  isch3  28082  19.9d2rf  29302  volfiniune  30278  bnj594  30967  bnj1371  31082  bnj1374  31084  dfrdg4  32042  bj-0nelsngl  32943  bj-ccinftydisj  33080  poimirlem25  33414  mblfinlem3  33428  mblfinlem4  33429  clsk3nimkb  38164  stoweidlem57  40043
  Copyright terms: Public domain W3C validator