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Theorem rabsneu 3443
 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 2315 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
21eqeq1i 2047 . . 3 ({𝑥𝐵𝜑} = {𝐴} ↔ {𝑥 ∣ (𝑥𝐵𝜑)} = {𝐴})
3 absneu 3442 . . 3 ((𝐴𝑉 ∧ {𝑥 ∣ (𝑥𝐵𝜑)} = {𝐴}) → ∃!𝑥(𝑥𝐵𝜑))
42, 3sylan2b 271 . 2 ((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥(𝑥𝐵𝜑))
5 df-reu 2313 . 2 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
64, 5sylibr 137 1 ((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥𝐵 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∃!weu 1900  {cab 2026  ∃!wreu 2308  {crab 2310  {csn 3375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-reu 2313  df-rab 2315  df-v 2559  df-sn 3381 This theorem is referenced by: (None)
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