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Theorem rexsngf 4594
 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 4593 . 2 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
2 rexsngf.1 . . 3 𝑥𝜓
3 rexsngf.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3sbciegf 3795 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
51, 4syl5bb 286 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2115  ∃wrex 3134  [wsbc 3758  {csn 4549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rex 3139  df-v 3482  df-sbc 3759  df-sn 4550 This theorem is referenced by:  reusngf  4596  rexsng  4598  rexprgf  4615  rmosn  4639  iunxsngf  5000
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