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Theorem reuhyp 4322
 Description: A theorem useful for eliminating restricted existential uniqueness hypotheses. (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 1300 . 2
2 reuhyp.1 . . . 4 (𝑥𝐶𝐵𝐶)
32adantl 272 . . 3 ((⊤ ∧ 𝑥𝐶) → 𝐵𝐶)
4 reuhyp.2 . . . 4 ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
543adant1 964 . . 3 ((⊤ ∧ 𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))
63, 5reuhypd 4321 . 2 ((⊤ ∧ 𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)
71, 6mpan 416 1 (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1296  ⊤wtru 1297   ∈ wcel 1445  ∃!wreu 2372 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-reu 2377  df-v 2635 This theorem is referenced by: (None)
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