Theorem spcev 2784

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 2777	. 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 1332   E.wex 1469   e. wcel 1481   _Vcvv 2689

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691

This theorem is referenced by:  bnd2  4105  mss  4156  exss  4157  snnex  4377  opeldm  4750  elrnmpt1  4798  xpmlem  4967  ffoss  5407  ssimaex  5490  fvelrn  5559  eufnfv  5656  foeqcnvco  5699  cnvoprab  6139  domtr  6687  ensn1  6698  ac6sfi  6800  difinfsn  6993  0ct  7000  ctmlemr  7001  ctssdclemn0  7003  ctssdclemr  7005  ctssdc  7006  omct  7010  ctssexmid  7032  exmidfodomrlemim  7074  cc3  7100  zfz1iso  10616  ennnfonelemim  11973  ctinfom  11977  ctinf  11979  qnnen  11980  enctlem  11981  ctiunct  11989  subctctexmid  13369