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Theorem reuhyp 5001
Description: A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4999. (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 1600 . 2
2 reuhyp.1 . . . 4 (𝑥𝐶𝐵𝐶)
32adantl 473 . . 3 ((⊤ ∧ 𝑥𝐶) → 𝐵𝐶)
4 reuhyp.2 . . . 4 ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
543adant1 1122 . . 3 ((⊤ ∧ 𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
63, 5reuhypd 5000 . 2 ((⊤ ∧ 𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
71, 6mpan 708 1 (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wtru 1597  wcel 2103  ∃!wreu 3016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-reu 3021  df-v 3306
This theorem is referenced by:  riotaneg  11115  zriotaneg  11604  zmax  11899  rebtwnz  11901
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