Theorem spcev 2825

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 2818	. 2 |- ( A e. _V -> ( ps -> E. x ph ) )
4	1, 3	ax-mp 5	1 |- ( ps -> E. x ph )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  bnd2  4157  mss  4209  exss  4210  snnex  4431  opeldm  4812  elrnmpt1  4860  xpmlem  5029  ffoss  5472  ssimaex  5555  fvelrn  5624  eufnfv  5723  foeqcnvco  5766  cnvoprab  6210  domtr  6759  ensn1  6770  ac6sfi  6872  difinfsn  7073  0ct  7080  ctmlemr  7081  ctssdclemn0  7083  ctssdclemr  7085  ctssdc  7086  omct  7090  ctssexmid  7122  exmidfodomrlemim  7165  cc3  7217  zfz1iso  10763  fprodntrivap  11534  ennnfonelemim  12366  ctinfom  12370  ctinf  12372  qnnen  12373  enctlem  12374  ctiunct  12382  nninfdc  12395  subctctexmid  13956