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Theorem rexsngf 40036
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 4436 . 2 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
2 rexsngf.1 . . 3 𝑥𝜓
3 rexsngf.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3sbciegf 3693 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
51, 4syl5bb 275 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1658  wnf 1884  wcel 2166  wrex 3117  [wsbc 3661  {csn 4396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2802
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-rex 3122  df-v 3415  df-sbc 3662  df-sn 4397
This theorem is referenced by:  iunxsngf2  40046
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