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Theorem rabsneu 3434
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 ((A 𝑉 {x Bφ} = {A}) → ∃!x B φ)

Proof of Theorem rabsneu
StepHypRef Expression
1 df-rab 2309 . . . 4 {x Bφ} = {x ∣ (x B φ)}
21eqeq1i 2044 . . 3 ({x Bφ} = {A} ↔ {x ∣ (x B φ)} = {A})
3 absneu 3433 . . 3 ((A 𝑉 {x ∣ (x B φ)} = {A}) → ∃!x(x B φ))
42, 3sylan2b 271 . 2 ((A 𝑉 {x Bφ} = {A}) → ∃!x(x B φ))
5 df-reu 2307 . 2 (∃!x B φ∃!x(x B φ))
64, 5sylibr 137 1 ((A 𝑉 {x Bφ} = {A}) → ∃!x B φ)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  ∃!weu 1897  {cab 2023  ∃!wreu 2302  {crab 2304  {csn 3367
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-reu 2307  df-rab 2309  df-v 2553  df-sn 3373
This theorem is referenced by: (None)
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