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Theorem rabsneu 4695
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 3411 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
21eqeq1i 2742 . . 3 ({𝑥𝐵𝜑} = {𝐴} ↔ {𝑥 ∣ (𝑥𝐵𝜑)} = {𝐴})
3 absneu 4694 . . 3 ((𝐴𝑉 ∧ {𝑥 ∣ (𝑥𝐵𝜑)} = {𝐴}) → ∃!𝑥(𝑥𝐵𝜑))
42, 3sylan2b 595 . 2 ((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥(𝑥𝐵𝜑))
5 df-reu 3357 . 2 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
64, 5sylibr 233 1 ((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  ∃!weu 2567  {cab 2714  ∃!wreu 3354  {crab 3410  {csn 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-reu 3357  df-rab 3411  df-sn 4592
This theorem is referenced by: (None)
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