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Theorem rabsneu 3680
Description: Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)
Assertion
Ref Expression
rabsneu ((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥𝐵 𝜑)

Proof of Theorem rabsneu
StepHypRef Expression
1 df-rab 2477 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
21eqeq1i 2197 . . 3 ({𝑥𝐵𝜑} = {𝐴} ↔ {𝑥 ∣ (𝑥𝐵𝜑)} = {𝐴})
3 absneu 3679 . . 3 ((𝐴𝑉 ∧ {𝑥 ∣ (𝑥𝐵𝜑)} = {𝐴}) → ∃!𝑥(𝑥𝐵𝜑))
42, 3sylan2b 287 . 2 ((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥(𝑥𝐵𝜑))
5 df-reu 2475 . 2 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
64, 5sylibr 134 1 ((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥𝐵 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  ∃!weu 2038  wcel 2160  {cab 2175  ∃!wreu 2470  {crab 2472  {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-reu 2475  df-rab 2477  df-v 2754  df-sn 3613
This theorem is referenced by: (None)
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