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Theorem rexsn 3768
 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 3766 . 2 (A V → (x {A}φψ))
41, 3ax-mp 8 1 (x {A}φψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-sbc 3047  df-sn 3741 This theorem is referenced by:  pw1sn  4165  elimaksn  4283  setswith  4321  addcid1  4405  ltfintrilem1  4465  nnadjoin  4520  tfinnn  4534  elsnres  4996  xpnedisj  5513  snec  5987  lec0cg  6198  addccan2nclem1  6263
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