MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rmoi2 Structured version   Visualization version   GIF version

Theorem rmoi2 3661
Description: Consequence of "restricted at most one." (Contributed by Thierry Arnoux, 9-Dec-2019.)
Hypotheses
Ref Expression
rmoi2.1 (𝑥 = 𝐵 → (𝜓𝜒))
rmoi2.2 (𝜑𝐵𝐴)
rmoi2.3 (𝜑 → ∃*𝑥𝐴 𝜓)
rmoi2.4 (𝜑𝑥𝐴)
rmoi2.5 (𝜑𝜓)
rmoi2.6 (𝜑𝜒)
Assertion
Ref Expression
rmoi2 (𝜑𝑥 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜒,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem rmoi2
StepHypRef Expression
1 rmoi2.6 . 2 (𝜑𝜒)
2 rmoi2.1 . . 3 (𝑥 = 𝐵 → (𝜓𝜒))
3 rmoi2.2 . . 3 (𝜑𝐵𝐴)
4 rmoi2.3 . . 3 (𝜑 → ∃*𝑥𝐴 𝜓)
5 rmoi2.4 . . 3 (𝜑𝑥𝐴)
6 rmoi2.5 . . 3 (𝜑𝜓)
72, 3, 4, 5, 6rmob2 3660 . 2 (𝜑 → (𝑥 = 𝐵𝜒))
81, 7mpbird 247 1 (𝜑𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1620  wcel 2127  ∃*wrmo 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-rmo 3046  df-v 3330
This theorem is referenced by:  lmieu  25846
  Copyright terms: Public domain W3C validator