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

Theorem rmoan 3556
Description: Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)
Assertion
Ref Expression
rmoan (∃*𝑥𝐴 𝜑 → ∃*𝑥𝐴 (𝜓𝜑))

Proof of Theorem rmoan
StepHypRef Expression
1 moan 2672 . . 3 (∃*𝑥(𝑥𝐴𝜑) → ∃*𝑥(𝜓 ∧ (𝑥𝐴𝜑)))
2 an12 616 . . . 4 ((𝜓 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝜓𝜑)))
32mobii 2640 . . 3 (∃*𝑥(𝜓 ∧ (𝑥𝐴𝜑)) ↔ ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
41, 3sylib 208 . 2 (∃*𝑥(𝑥𝐴𝜑) → ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
5 df-rmo 3068 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
6 df-rmo 3068 . 2 (∃*𝑥𝐴 (𝜓𝜑) ↔ ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
74, 5, 63imtr4i 281 1 (∃*𝑥𝐴 𝜑 → ∃*𝑥𝐴 (𝜓𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2144  ∃*wmo 2618  ∃*wrmo 3063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-12 2202
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-eu 2621  df-mo 2622  df-rmo 3068
This theorem is referenced by:  reuxfr2d  5019  reuxfr3d  29662
  Copyright terms: Public domain W3C validator