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

Theorem rmoan 3681
 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 2614 . . 3 (∃*𝑥(𝑥𝐴𝜑) → ∃*𝑥(𝜓 ∧ (𝑥𝐴𝜑)))
2 an12 644 . . . 4 ((𝜓 ∧ (𝑥𝐴𝜑)) ↔ (𝑥𝐴 ∧ (𝜓𝜑)))
32mobii 2609 . . 3 (∃*𝑥(𝜓 ∧ (𝑥𝐴𝜑)) ↔ ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
41, 3sylib 221 . 2 (∃*𝑥(𝑥𝐴𝜑) → ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
5 df-rmo 3117 . 2 (∃*𝑥𝐴 𝜑 ↔ ∃*𝑥(𝑥𝐴𝜑))
6 df-rmo 3117 . 2 (∃*𝑥𝐴 (𝜓𝜑) ↔ ∃*𝑥(𝑥𝐴 ∧ (𝜓𝜑)))
74, 5, 63imtr4i 295 1 (∃*𝑥𝐴 𝜑 → ∃*𝑥𝐴 (𝜓𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2112  ∃*wmo 2599  ∃*wrmo 3112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-mo 2601  df-rmo 3117 This theorem is referenced by:  reuxfrd  3690  reuxfrdf  30266
 Copyright terms: Public domain W3C validator