Theorem spcv 2911

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 2904	. 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 1396   = wceq 1398   e. wcel 2203   _Vcvv 2813

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815

This theorem is referenced by:  morex  3001  exmidexmid  4309  exmidsssn  4315  exmidel  4318  rext  4331  ontr2exmid  4647  regexmidlem1  4655  reg2exmid  4658  relop  4905  uchoice  6331  disjxp1  6432  rdgtfr  6605  ssfiexmid  7131  ssfiexmidt  7133  domfiexmid  7135  diffitest  7144  findcard  7145  exmidpw2en  7172  fiintim  7191  fisseneq  7195  finomni  7431  exmidomni  7433  exmidlpo  7434  ballotfilem2  13142  exmidunben  13177  ivthreinc  15510  bj-d0clsepcl  16695  bj-inf2vnlem1  16740  subctctexmid  16774