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Theorem rabsneu 3795
 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 B φ} = {A}) → ∃!x B φ)

Proof of Theorem rabsneu
StepHypRef Expression
1 df-rab 2623 . . . 4 {x B φ} = {x (x B φ)}
21eqeq1i 2360 . . 3 ({x B φ} = {A} ↔ {x (x B φ)} = {A})
3 absneu 3794 . . 3 ((A V {x (x B φ)} = {A}) → ∃!x(x B φ))
42, 3sylan2b 461 . 2 ((A V {x B φ} = {A}) → ∃!x(x B φ))
5 df-reu 2621 . 2 (∃!x B φ∃!x(x B φ))
64, 5sylibr 203 1 ((A V {x B φ} = {A}) → ∃!x B φ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  {cab 2339  ∃!wreu 2616  {crab 2618  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-reu 2621  df-rab 2623  df-v 2861  df-sn 3741 This theorem is referenced by: (None)
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