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Theorem rexsngf 4677
Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.) (Revised 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 4676 . 2 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
2 rexsngf.1 . . 3 𝑥𝜓
3 rexsngf.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3sbciegf 3831 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
51, 4bitrid 283 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wnf 1780  wcel 2106  wrex 3068  [wsbc 3791  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-v 3480  df-sbc 3792  df-sn 4632
This theorem is referenced by:  reusngf  4679  rexprgf  4700  rmosn  4724  iunxsngf  5097
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