Theorem spcev 2912

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 2905	. 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 1398   E.wex 1541   e. wcel 2203   _Vcvv 2813

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815

This theorem is referenced by:  bnd2  4286  mss  4342  exss  4343  snnex  4569  opeldm  4959  elrnmpt1  5008  xpmlem  5183  ffoss  5647  ssimaex  5738  fvelrn  5808  funopsn  5860  eufnfv  5917  foeqcnvco  5963  cnvoprab  6430  domtr  7025  ensn1  7036  ac6sfi  7155  difinfsn  7391  0ct  7398  ctmlemr  7399  ctssdclemn0  7401  ctssdclemr  7403  ctssdc  7404  omct  7408  ctssexmid  7441  exmidfodomrlemim  7504  cc3  7582  zfz1iso  11213  fzf1o  12061  fprodntrivap  12270  nninfct  12737  ennnfonelemim  13175  ctinfom  13179  ctinf  13181  qnnen  13182  enctlem  13183  ctiunct  13191  nninfdc  13204  subctctexmid  16774  domomsubct  16775