MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabsneu Structured version   Visualization version   GIF version

Theorem rabsneu 4578
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 3116 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
21eqeq1i 2802 . . 3 ({𝑥𝐵𝜑} = {𝐴} ↔ {𝑥 ∣ (𝑥𝐵𝜑)} = {𝐴})
3 absneu 4577 . . 3 ((𝐴𝑉 ∧ {𝑥 ∣ (𝑥𝐵𝜑)} = {𝐴}) → ∃!𝑥(𝑥𝐵𝜑))
42, 3sylan2b 593 . 2 ((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥(𝑥𝐵𝜑))
5 df-reu 3114 . 2 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
64, 5sylibr 235 1 ((𝐴𝑉 ∧ {𝑥𝐵𝜑} = {𝐴}) → ∃!𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  wcel 2083  ∃!weu 2613  {cab 2777  ∃!wreu 3109  {crab 3111  {csn 4478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-reu 3114  df-rab 3116  df-sn 4479
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator