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Theorem rexsns 4374
Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)
Assertion
Ref Expression
rexsns (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexsns
StepHypRef Expression
1 velsn 4350 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21anbi1i 617 . . 3 ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ (𝑥 = 𝐴𝜑))
32exbii 1943 . 2 (∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑))
4 df-rex 3061 . 2 (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑))
5 sbc5 3621 . 2 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
63, 4, 53bitr4i 294 1 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wrex 3056  [wsbc 3596  {csn 4334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-v 3352  df-sbc 3597  df-sn 4335
This theorem is referenced by:  rexsng  4376  r19.12sn  4410  iunxsngf  4760  poimirlem25  33790  rexsngf  39803
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