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Theorem spcev 2833
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 2826 . 2 (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
41, 3ax-mp 5 1 (𝜓 → ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wex 1492  wcel 2148  Vcvv 2738
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 2740
This theorem is referenced by:  bnd2  4174  mss  4227  exss  4228  snnex  4449  opeldm  4831  elrnmpt1  4879  xpmlem  5050  ffoss  5494  ssimaex  5578  fvelrn  5648  eufnfv  5748  foeqcnvco  5791  cnvoprab  6235  domtr  6785  ensn1  6796  ac6sfi  6898  difinfsn  7099  0ct  7106  ctmlemr  7107  ctssdclemn0  7109  ctssdclemr  7111  ctssdc  7112  omct  7116  ctssexmid  7148  exmidfodomrlemim  7200  cc3  7267  zfz1iso  10821  fprodntrivap  11592  ennnfonelemim  12425  ctinfom  12429  ctinf  12431  qnnen  12432  enctlem  12433  ctiunct  12441  nninfdc  12454  subctctexmid  14753
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