Theorem spcv 2900

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 2893	. 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 1395   = wceq 1397   e. wcel 2202   _Vcvv 2802

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804

This theorem is referenced by:  morex  2990  exmidexmid  4286  exmidsssn  4292  exmidel  4295  rext  4307  ontr2exmid  4623  regexmidlem1  4631  reg2exmid  4634  relop  4880  uchoice  6299  disjxp1  6400  rdgtfr  6539  ssfiexmid  7062  ssfiexmidt  7064  domfiexmid  7066  diffitest  7075  findcard  7076  exmidpw2en  7103  fiintim  7122  fisseneq  7126  finomni  7338  exmidomni  7340  exmidlpo  7341  exmidunben  13046  ivthreinc  15368  bj-d0clsepcl  16520  bj-inf2vnlem1  16565  subctctexmid  16601