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Theorem elabd 2884
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 2827 . 2 (𝑋 ∈ V → (𝜒 → ∃𝑥𝜓))
51, 2, 4sylc 62 1 (𝜑 → ∃𝑥𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wex 1492  wcel 2148  Vcvv 2739
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-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741
This theorem is referenced by:  ntrivcvgap0  11559  ssomct  12448  dceqnconst  14847  dcapnconst  14848
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