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Theorem rexsns 3480
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 3461 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21anbi1i 446 . . 3 ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ (𝑥 = 𝐴𝜑))
32exbii 1541 . 2 (∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴𝜑))
4 df-rex 2365 . 2 (∃𝑥 ∈ {𝐴}𝜑 ↔ ∃𝑥(𝑥 ∈ {𝐴} ∧ 𝜑))
5 sbc5 2863 . 2 ([𝐴 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝐴𝜑))
63, 4, 53bitr4i 210 1 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  wex 1426  wcel 1438  wrex 2360  [wsbc 2840  {csn 3444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-sbc 2841  df-sn 3450
This theorem is referenced by:  rexsng  3482  r19.12sn  3506  iunxsngf  3805  finexdc  6608  exfzdc  9639
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