Theorem spcev 2834

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 2827	. 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 1353   E.wex 1492   e. wcel 2148   _Vcvv 2739

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 2741

This theorem is referenced by:  bnd2  4175  mss  4228  exss  4229  snnex  4450  opeldm  4832  elrnmpt1  4880  xpmlem  5051  ffoss  5495  ssimaex  5579  fvelrn  5649  eufnfv  5749  foeqcnvco  5793  cnvoprab  6237  domtr  6787  ensn1  6798  ac6sfi  6900  difinfsn  7101  0ct  7108  ctmlemr  7109  ctssdclemn0  7111  ctssdclemr  7113  ctssdc  7114  omct  7118  ctssexmid  7150  exmidfodomrlemim  7202  cc3  7269  zfz1iso  10823  fprodntrivap  11594  ennnfonelemim  12427  ctinfom  12431  ctinf  12433  qnnen  12434  enctlem  12435  ctiunct  12443  nninfdc  12456  subctctexmid  14835