Theorem spcev 2898

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 2891	. 2 |- ( A e. _V -> ( ps -> E. x ph ) )
4	1, 3	ax-mp 5	1 |- ( ps -> E. x ph )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  bnd2  4257  mss  4312  exss  4313  snnex  4539  opeldm  4926  elrnmpt1  4975  xpmlem  5149  ffoss  5604  ssimaex  5695  fvelrn  5766  funopsn  5817  eufnfv  5870  foeqcnvco  5914  cnvoprab  6380  domtr  6937  ensn1  6948  ac6sfi  7060  difinfsn  7267  0ct  7274  ctmlemr  7275  ctssdclemn0  7277  ctssdclemr  7279  ctssdc  7280  omct  7284  ctssexmid  7317  exmidfodomrlemim  7379  cc3  7454  zfz1iso  11063  fprodntrivap  12095  nninfct  12562  ennnfonelemim  12995  ctinfom  12999  ctinf  13001  qnnen  13002  enctlem  13003  ctiunct  13011  nninfdc  13024  subctctexmid  16366  domomsubct  16367