Theorem spcev 2875

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 2868	. 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 1373   E.wex 1516   e. wcel 2178   _Vcvv 2776

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778

This theorem is referenced by:  bnd2  4233  mss  4288  exss  4289  snnex  4513  opeldm  4900  elrnmpt1  4948  xpmlem  5122  ffoss  5576  ssimaex  5663  fvelrn  5734  funopsn  5785  eufnfv  5838  foeqcnvco  5882  cnvoprab  6343  domtr  6900  ensn1  6911  ac6sfi  7021  difinfsn  7228  0ct  7235  ctmlemr  7236  ctssdclemn0  7238  ctssdclemr  7240  ctssdc  7241  omct  7245  ctssexmid  7278  exmidfodomrlemim  7340  cc3  7415  zfz1iso  11023  fprodntrivap  12010  nninfct  12477  ennnfonelemim  12910  ctinfom  12914  ctinf  12916  qnnen  12917  enctlem  12918  ctiunct  12926  nninfdc  12939  subctctexmid  16139  domomsubct  16140