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

Theorem rmoan 3388
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 2523 . . 3 (∃*𝑥(𝑥𝐴𝜑) → ∃*𝑥(𝜓 ∧ (𝑥𝐴𝜑)))
2 an12 837 . . . 4 ((𝜓 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝜓𝜑)))
32mobii 2492 . . 3 (∃*𝑥(𝜓 ∧ (𝑥𝐴𝜑)) ↔ ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
41, 3sylib 208 . 2 (∃*𝑥(𝑥𝐴𝜑) → ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
5 df-rmo 2915 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
6 df-rmo 2915 . 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 384  wcel 1987  ∃*wmo 2470  ∃*wrmo 2910
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 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
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 2473  df-mo 2474  df-rmo 2915
This theorem is referenced by:  reuxfr2d  4851  reuxfr3d  29175
  Copyright terms: Public domain W3C validator