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

Theorem 2moexv 2711
Description: Double quantification with "at most one". (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2moexv (∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2moexv
StepHypRef Expression
1 nfe1 2147 . . 3 𝑦𝑦𝜑
21nfmov 2642 . 2 𝑦∃*𝑥𝑦𝜑
3 19.8a 2172 . . 3 (𝜑 → ∃𝑦𝜑)
43moimi 2625 . 2 (∃*𝑥𝑦𝜑 → ∃*𝑥𝜑)
52, 4alrimi 2206 1 (∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1528  wex 1773  ∃*wmo 2618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-11 2153  ax-12 2169
This theorem depends on definitions:  df-bi 208  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-mo 2620
This theorem is referenced by:  2eu5  2741
  Copyright terms: Public domain W3C validator