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  4218  mss  4271  exss  4272  snnex  4496  opeldm  4882  elrnmpt1  4930  xpmlem  5104  ffoss  5556  ssimaex  5642  fvelrn  5713  funopsn  5764  eufnfv  5817  foeqcnvco  5861  cnvoprab  6322  domtr  6879  ensn1  6890  ac6sfi  6997  difinfsn  7204  0ct  7211  ctmlemr  7212  ctssdclemn0  7214  ctssdclemr  7216  ctssdc  7217  omct  7221  ctssexmid  7254  exmidfodomrlemim  7311  cc3  7382  zfz1iso  10988  fprodntrivap  11928  nninfct  12395  ennnfonelemim  12828  ctinfom  12832  ctinf  12834  qnnen  12835  enctlem  12836  ctiunct  12844  nninfdc  12857  subctctexmid  15974  domomsubct  15975