Theorem spcev 2713

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 2707	. 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 1289   E.wex 1426   e. wcel 1438   _Vcvv 2619

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621

This theorem is referenced by:  bnd2  4006  mss  4051  exss  4052  snnex  4269  opeldm  4635  elrnmpt1  4682  xpmlem  4847  ffoss  5279  ssimaex  5359  fvelrn  5424  eufnfv  5517  foeqcnvco  5561  cnvoprab  5991  domtr  6492  ensn1  6503  ac6sfi  6604  exmidfodomrlemim  6817  zfz1iso  10234