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Theorem rexsns 4666
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 4637 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21anbi1i 623 . . 3 ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ (𝑥 = 𝐴𝜑))
32exbii 1842 . 2 (∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑))
4 df-rex 3063 . 2 (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑))
5 sbc5 3798 . 2 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
63, 4, 53bitr4i 303 1 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  wrex 3062  [wsbc 3770  {csn 4621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rex 3063  df-v 3468  df-sbc 3771  df-sn 4622
This theorem is referenced by:  rexsngf  4667  r19.12sn  4717  poimirlem25  37017
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