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

Theorem moanmo 2520
 Description: Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)
Assertion
Ref Expression
moanmo ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)

Proof of Theorem moanmo
StepHypRef Expression
1 id 22 . . 3 (∃*𝑥𝜑 → ∃*𝑥𝜑)
2 nfmo1 2469 . . . 4 𝑥∃*𝑥𝜑
32moanim 2517 . . 3 (∃*𝑥(∃*𝑥𝜑𝜑) ↔ (∃*𝑥𝜑 → ∃*𝑥𝜑))
41, 3mpbir 220 . 2 ∃*𝑥(∃*𝑥𝜑𝜑)
5 ancom 465 . . 3 ((𝜑 ∧ ∃*𝑥𝜑) ↔ (∃*𝑥𝜑𝜑))
65mobii 2481 . 2 (∃*𝑥(𝜑 ∧ ∃*𝑥𝜑) ↔ ∃*𝑥(∃*𝑥𝜑𝜑))
74, 6mpbir 220 1 ∃*𝑥(𝜑 ∧ ∃*𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∃*wmo 2459 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463 This theorem is referenced by:  moaneu  2521
 Copyright terms: Public domain W3C validator