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Theorem reuhyp 5298
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr1 3718. (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 1542 . 2
2 reuhyp.1 . . . 4 (𝑥𝐶𝐵𝐶)
32adantl 485 . . 3 ((⊤ ∧ 𝑥𝐶) → 𝐵𝐶)
4 reuhyp.2 . . . 4 ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
543adant1 1127 . . 3 ((⊤ ∧ 𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
63, 5reuhypd 5297 . 2 ((⊤ ∧ 𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
71, 6mpan 689 1 (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wtru 1539  wcel 2114  ∃!wreu 3132
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-reu 3137  df-v 3471
This theorem is referenced by:  riotaneg  11607  zriotaneg  12084  zmax  12333  rebtwnz  12335
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