Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rexsngf Structured version   Visualization version   GIF version

Theorem rexsngf 39046
Description: Restricted existential quantification over a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
rexsngf.1 𝑥𝜓
rexsngf.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rexsngf (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem rexsngf
StepHypRef Expression
1 rexsns 4215 . 2 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
2 rexsngf.1 . . 3 𝑥𝜓
3 rexsngf.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3sbciegf 3465 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
51, 4syl5bb 272 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1482  wnf 1707  wcel 1989  wrex 2912  [wsbc 3433  {csn 4175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rex 2917  df-v 3200  df-sbc 3434  df-sn 4176
This theorem is referenced by:  iunxsngf2  39056
  Copyright terms: Public domain W3C validator