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Theorem elabd 2829
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 2774 . 2 (𝑋 ∈ V → (𝜒 → ∃𝑥𝜓))
51, 2, 4sylc 62 1 (𝜑 → ∃𝑥𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wex 1468  wcel 1480  Vcvv 2686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688
This theorem is referenced by:  ntrivcvgap0  11342  dcapncf  13396
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