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Theorem elabd 3542
Description: Explicit demonstration the class {𝑥𝜓} is not empty by the example 𝑋. (Contributed by RP, 12-Aug-2020.)
Hypotheses
Ref Expression
elab.xex (𝜑𝑋 ∈ V)
elab.xmaj (𝜑𝜒)
elab.xsub (𝑥 = 𝑋 → (𝜓𝜒))
Assertion
Ref Expression
elabd (𝜑 → ∃𝑥𝜓)
Distinct variable groups:   𝜒,𝑥   𝑥,𝑋
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem elabd
StepHypRef Expression
1 elab.xex . 2 (𝜑𝑋 ∈ V)
2 elab.xmaj . 2 (𝜑𝜒)
3 elab.xsub . . 3 (𝑥 = 𝑋 → (𝜓𝜒))
43spcegv 3480 . 2 (𝑋 ∈ V → (𝜒 → ∃𝑥𝜓))
51, 2, 4sylc 65 1 (𝜑 → ∃𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wex 1875  wcel 2157  Vcvv 3383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-v 3385
This theorem is referenced by:  hasheqf1od  13390  setsexstruct2  16220  wwlksnextbij  27171  wwlksnextbijOLD  27172  eqvreltr  34835  clrellem  38700  clcnvlem  38701  isomgreqve  42483  isomushgr  42484  isomgrsym  42494  isomgrtr  42497  ushrisomgr  42499  uspgrsprfo  42543  uspgrbispr  42546
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