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Theorem rexsn 4362
 Description: Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)
Hypotheses
Ref Expression
ralsn.1 𝐴 ∈ V
ralsn.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rexsn (∃𝑥 ∈ {𝐴}𝜑𝜓)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexsn
StepHypRef Expression
1 ralsn.1 . 2 𝐴 ∈ V
2 ralsn.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32rexsng 4358 . 2 (𝐴 ∈ V → (∃𝑥 ∈ {𝐴}𝜑𝜓))
41, 3ax-mp 5 1 (∃𝑥 ∈ {𝐴}𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  Vcvv 3351  {csn 4317 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-v 3353  df-sbc 3588  df-sn 4318 This theorem is referenced by:  elsnres  5576  oarec  7800  snec  7966  zornn0g  9533  fpwwe2lem13  9670  elreal  10158  hashge2el2difr  13465  vdwlem6  15897  pmatcollpw3fi1  20813  restsn  21195  snclseqg  22139  ust0  22243  esum2dlem  30494  eulerpartlemgh  30780  eldm3  31989  poimirlem28  33770  heiborlem3  33944
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