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

Theorem rexex 2985
 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 2902 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 exsimpr 1784 . 2 (∃𝑥(𝑥𝐴𝜑) → ∃𝑥𝜑)
31, 2sylbi 206 1 (∃𝑥𝐴 𝜑 → ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-rex 2902 This theorem is referenced by:  reu3  3363  rmo2i  3493  dffo5  6284  nqerf  9631  supsrlem  9811  vdwmc2  15521  isch3  27482  19.9d2rf  28702  volfiniune  29620  bnj594  30236  bnj1371  30351  bnj1374  30353  dfrdg4  31228  bj-0nelsngl  32152  bj-toprntopon  32244  bj-ccinftydisj  32277  poimirlem25  32604  mblfinlem3  32618  mblfinlem4  32619  clsk3nimkb  37358  stoweidlem57  38950
 Copyright terms: Public domain W3C validator