MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2moex Structured version   Visualization version   GIF version

Theorem 2moex 2547
Description: Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2moex (∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝜑)

Proof of Theorem 2moex
StepHypRef Expression
1 nfe1 2029 . . 3 𝑦𝑦𝜑
21nfmo 2491 . 2 𝑦∃*𝑥𝑦𝜑
3 19.8a 2054 . . 3 (𝜑 → ∃𝑦𝜑)
43moimi 2524 . 2 (∃*𝑥𝑦𝜑 → ∃*𝑥𝜑)
52, 4alrimi 2085 1 (∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wex 1701  ∃*wmo 2475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-eu 2478  df-mo 2479
This theorem is referenced by:  2eu2  2558  2eu5  2561
  Copyright terms: Public domain W3C validator