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Theorem reu6i 3737
Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
reu6i ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reu6i
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2747 . . . . 5 (𝑦 = 𝐵 → (𝑥 = 𝑦𝑥 = 𝐵))
21bibi2d 342 . . . 4 (𝑦 = 𝐵 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐵)))
32ralbidv 3176 . . 3 (𝑦 = 𝐵 → (∀𝑥𝐴 (𝜑𝑥 = 𝑦) ↔ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)))
43rspcev 3622 . 2 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
5 reu6 3735 . 2 (∃!𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝜑𝑥 = 𝑦))
64, 5sylibr 234 1 ((𝐵𝐴 ∧ ∀𝑥𝐴 (𝜑𝑥 = 𝐵)) → ∃!𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  ∃!wreu 3376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-reu 3379
This theorem is referenced by:  eqreu  3738  riota5f  7416  negeu  11496  creur  12258  creui  12259  reuccatpfxs1  14782  lublecl  18419  dfod2  19597  lmieu  28807  esum2dlem  34073  fvineqsneu  37394  poimirlem16  37623  poimirlem17  37624  poimirlem19  37626  poimirlem20  37627  poimirlem22  37629  renegeulemv  42375  sn-subeu  42433  upeu2lem  48808
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