Theorem spcev 2868

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 2861	. 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 1373   E.wex 1515   e. wcel 2176   _Vcvv 2772

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774

This theorem is referenced by:  bnd2  4217  mss  4270  exss  4271  snnex  4495  opeldm  4881  elrnmpt1  4929  xpmlem  5103  ffoss  5554  ssimaex  5640  fvelrn  5711  funopsn  5762  eufnfv  5815  foeqcnvco  5859  cnvoprab  6320  domtr  6877  ensn1  6888  ac6sfi  6995  difinfsn  7202  0ct  7209  ctmlemr  7210  ctssdclemn0  7212  ctssdclemr  7214  ctssdc  7215  omct  7219  ctssexmid  7252  exmidfodomrlemim  7309  cc3  7380  zfz1iso  10986  fprodntrivap  11895  nninfct  12362  ennnfonelemim  12795  ctinfom  12799  ctinf  12801  qnnen  12802  enctlem  12803  ctiunct  12811  nninfdc  12824  subctctexmid  15941  domomsubct  15942