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Theorem reuhyp 5137
 Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 5135. (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 1606 . 2
2 reuhyp.1 . . . 4 (𝑥𝐶𝐵𝐶)
32adantl 475 . . 3 ((⊤ ∧ 𝑥𝐶) → 𝐵𝐶)
4 reuhyp.2 . . . 4 ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
543adant1 1121 . . 3 ((⊤ ∧ 𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
63, 5reuhypd 5136 . 2 ((⊤ ∧ 𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
71, 6mpan 680 1 (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601  ⊤wtru 1602   ∈ wcel 2107  ∃!wreu 3092 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-reu 3097  df-v 3400 This theorem is referenced by:  riotaneg  11360  zriotaneg  11847  zmax  12096  rebtwnz  12098
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