Theorem spcev 2692

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 2686	. 2 |- ( A e. _V -> ( ps -> E. x ph ) )
4	1, 3	ax-mp 7	1 |- ( ps -> E. x ph )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  bnd2  3947  mss  3981  exss  3982  snnex  4199  opeldm  4556  elrnmpt1  4603  xpmlem  4764  ffoss  5178  ssimaex  5255  fvelrn  5319  eufnfv  5410  foeqcnvco  5450  cnvoprab  5875  domtr  6288  ensn1  6299  ac6sfi  6379