Theorem spcev 2775

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 e. _V
spcv.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
spcev	|- ( ps -> E. x ph )

Distinct variable groups:   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem spcev

Step	Hyp	Ref	Expression
1		spcv.1	. 2 |- A e. _V
2		spcv.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	spcegv 2769	. 2 |- ( A e. _V -> ( ps -> E. x ph ) )
4	1, 3	ax-mp 5	1 |- ( ps -> E. x ph )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1331   E.wex 1468   e. wcel 1480   _Vcvv 2681

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683

This theorem is referenced by:  bnd2  4092  mss  4143  exss  4144  snnex  4364  opeldm  4737  elrnmpt1  4785  xpmlem  4954  ffoss  5392  ssimaex  5475  fvelrn  5544  eufnfv  5641  foeqcnvco  5684  cnvoprab  6124  domtr  6672  ensn1  6683  ac6sfi  6785  difinfsn  6978  0ct  6985  ctmlemr  6986  ctssdclemn0  6988  ctssdclemr  6990  ctssdc  6991  omct  6995  ctssexmid  7017  exmidfodomrlemim  7050  zfz1iso  10577  ennnfonelemim  11926  ctinfom  11930  ctinf  11932  qnnen  11933  enctlem  11934  ctiunct  11942  subctctexmid  13185