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Theorem rmoi2 3494
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 3493 . 2 (𝜑 → (𝑥 = 𝐵𝜒))
81, 7mpbird 245 1 (𝜑𝑥 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194   = wceq 1474  wcel 1976  ∃*wrmo 2895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-rmo 2900  df-v 3171
This theorem is referenced by:  lmieu  25391
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