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Theorem spcev 2713
Description: Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
Hypotheses
Ref Expression
spcv.1 𝐴 ∈ V
spcv.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
spcev (𝜓 → ∃𝑥𝜑)
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem spcev
StepHypRef Expression
1 spcv.1 . 2 𝐴 ∈ V
2 spcv.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
32spcegv 2707 . 2 (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
41, 3ax-mp 7 1 (𝜓 → ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  bnd2  4000  mss  4044  exss  4045  snnex  4262  opeldm  4627  elrnmpt1  4674  xpmlem  4839  ffoss  5269  ssimaex  5349  fvelrn  5414  eufnfv  5507  foeqcnvco  5551  cnvoprab  5981  domtr  6482  ensn1  6493  ac6sfi  6594  exmidfodomrlemim  6806  zfz1iso  10211
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