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Theorem rexsng 3633
Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
Hypothesis
Ref Expression
ralsng.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rexsng (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem rexsng
StepHypRef Expression
1 rexsns 3631 . 2 (∃𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
2 ralsng.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32sbcieg 2995 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝜑𝜓))
41, 3bitrid 192 1 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  wrex 2456  [wsbc 2962  {csn 3592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-sn 3598
This theorem is referenced by:  rexsn  3636  rexprg  3644  rextpg  3646  iunxsng  3962  imasng  4993  dvdsprmpweqnn  12329  mnd1  12841  grp1  12970
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