Theorem spcv 2898

Description: Rule of specialization, using implicit substitution. (Contributed by NM, 22-Jun-1994.)

Hypotheses

Ref	Expression
spcv.1	|- A e. _V
spcv.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
spcv	|- ( A. x ph -> ps )

Distinct variable groups:   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem spcv

Step	Hyp	Ref	Expression
1		spcv.1	. 2 |- A e. _V
2		spcv.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	spcgv 2891	. 2 |- ( A e. _V -> ( A. x ph -> ps ) )
4	1, 3	ax-mp 5	1 |- ( A. x ph -> ps )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   e. wcel 2200   _Vcvv 2800

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802

This theorem is referenced by:  morex  2988  exmidexmid  4284  exmidsssn  4290  exmidel  4293  rext  4305  ontr2exmid  4621  regexmidlem1  4629  reg2exmid  4632  relop  4878  uchoice  6295  disjxp1  6396  rdgtfr  6535  ssfiexmid  7058  domfiexmid  7060  diffitest  7069  findcard  7070  exmidpw2en  7097  fiintim  7116  fisseneq  7119  finomni  7330  exmidomni  7332  exmidlpo  7333  exmidunben  13037  ivthreinc  15359  bj-d0clsepcl  16456  bj-inf2vnlem1  16501  subctctexmid  16537