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Theorem spcev 2752
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 2746 . 2 (𝐴 ∈ V → (𝜓 → ∃𝑥𝜑))
41, 3ax-mp 5 1 (𝜓 → ∃𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wex 1451  wcel 1463  Vcvv 2658
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660
This theorem is referenced by:  bnd2  4065  mss  4116  exss  4117  snnex  4337  opeldm  4710  elrnmpt1  4758  xpmlem  4927  ffoss  5365  ssimaex  5448  fvelrn  5517  eufnfv  5614  foeqcnvco  5657  cnvoprab  6097  domtr  6645  ensn1  6656  ac6sfi  6758  difinfsn  6951  0ct  6958  ctmlemr  6959  ctssdclemn0  6961  ctssdclemr  6963  ctssdc  6964  omct  6968  ctssexmid  6990  exmidfodomrlemim  7021  zfz1iso  10524  ennnfonelemim  11832  ctinfom  11836  ctinf  11838  qnnen  11839  enctlem  11840  ctiunct  11848  subctctexmid  12998
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