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

Theorem moani 2674
 Description: "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
Hypothesis
Ref Expression
moani.1 ∃*𝑥𝜑
Assertion
Ref Expression
moani ∃*𝑥(𝜓𝜑)

Proof of Theorem moani
StepHypRef Expression
1 moani.1 . 2 ∃*𝑥𝜑
2 moan 2673 . 2 (∃*𝑥𝜑 → ∃*𝑥(𝜓𝜑))
31, 2ax-mp 5 1 ∃*𝑥(𝜓𝜑)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 382  ∃*wmo 2619 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-10 2174  ax-12 2203 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ex 1853  df-nf 1858  df-eu 2622  df-mo 2623 This theorem is referenced by:  euxfr2  3543  rmoeq  3557  reuxfr2d  5019  fvopab6  6453  1stconst  7416  2ndconst  7417  iunmapdisj  9046  axaddf  10168  axmulf  10169  joinval  17213  meetval  17227  reuxfr3d  29668
 Copyright terms: Public domain W3C validator