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Theorem rmoi2 3902
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 3901 . 2 (𝜑 → (𝑥 = 𝐵𝜒))
81, 7mpbird 257 1 (𝜑𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  ∃*wrmo 3377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-rmo 3378  df-v 3480
This theorem is referenced by:  lmieu  28807
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