Theorem spcev 2902

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 2895	. 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 2202   _Vcvv 2803

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805

This theorem is referenced by:  bnd2  4269  mss  4324  exss  4325  snnex  4551  opeldm  4940  elrnmpt1  4989  xpmlem  5164  ffoss  5625  ssimaex  5716  fvelrn  5786  funopsn  5838  eufnfv  5895  foeqcnvco  5941  cnvoprab  6408  domtr  7002  ensn1  7013  ac6sfi  7130  difinfsn  7359  0ct  7366  ctmlemr  7367  ctssdclemn0  7369  ctssdclemr  7371  ctssdc  7372  omct  7376  ctssexmid  7409  exmidfodomrlemim  7472  cc3  7547  zfz1iso  11168  fzf1o  12016  fprodntrivap  12225  nninfct  12692  ennnfonelemim  13125  ctinfom  13129  ctinf  13131  qnnen  13132  enctlem  13133  ctiunct  13141  nninfdc  13154  subctctexmid  16722  domomsubct  16723