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Theorem elabd 3501
 Description: Explicit demonstration the class {𝑥 ∣ 𝜓} 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 3443 . 2 (𝑋 ∈ V → (𝜒 → ∃𝑥𝜓))
51, 2, 4sylc 65 1 (𝜑 → ∃𝑥𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1630  ∃wex 1851   ∈ wcel 2144  Vcvv 3349 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351 This theorem is referenced by:  hasheqf1od  13345  setsexstruct2  16103  wwlksnextbij  27044  clrellem  38448  clcnvlem  38449  uspgrsprfo  42274  uspgrbispr  42277
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