ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  spcev GIF version

Theorem spcev 2832
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 2825 . 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 2737
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 2739
This theorem is referenced by:  bnd2  4173  mss  4226  exss  4227  snnex  4448  opeldm  4830  elrnmpt1  4878  xpmlem  5049  ffoss  5493  ssimaex  5577  fvelrn  5647  eufnfv  5747  foeqcnvco  5790  cnvoprab  6234  domtr  6784  ensn1  6795  ac6sfi  6897  difinfsn  7098  0ct  7105  ctmlemr  7106  ctssdclemn0  7108  ctssdclemr  7110  ctssdc  7111  omct  7115  ctssexmid  7147  exmidfodomrlemim  7199  cc3  7266  zfz1iso  10816  fprodntrivap  11587  ennnfonelemim  12419  ctinfom  12423  ctinf  12425  qnnen  12426  enctlem  12427  ctiunct  12435  nninfdc  12448  subctctexmid  14670
  Copyright terms: Public domain W3C validator