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Theorem rmoi2 3498
 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 3497 . 2 (𝜑 → (𝑥 = 𝐵𝜒))
81, 7mpbird 246 1 (𝜑𝑥 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∃*wrmo 2899 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  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rmo 2904  df-v 3175 This theorem is referenced by:  lmieu  25476
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