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Theorem reuhyp 5373
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr1 3708. (Contributed by NM, 15-Nov-2004.)
Hypotheses
Ref Expression
reuhyp.1 (𝑥𝐶𝐵𝐶)
reuhyp.2 ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
Assertion
Ref Expression
reuhyp (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)
Distinct variable groups:   𝑦,𝐵   𝑦,𝐶   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem reuhyp
StepHypRef Expression
1 tru 1545 . 2
2 reuhyp.1 . . . 4 (𝑥𝐶𝐵𝐶)
32adantl 482 . . 3 ((⊤ ∧ 𝑥𝐶) → 𝐵𝐶)
4 reuhyp.2 . . . 4 ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
543adant1 1130 . . 3 ((⊤ ∧ 𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
63, 5reuhypd 5372 . 2 ((⊤ ∧ 𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
71, 6mpan 688 1 (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wtru 1542  wcel 2106  ∃!wreu 3349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-reu 3352  df-v 3445
This theorem is referenced by:  riotaneg  12092  zriotaneg  12574  zmax  12824  rebtwnz  12826
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