ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  spcv Unicode version

Theorem spcv 2780
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
StepHypRef Expression
1 spcv.1 . 2  |-  A  e. 
_V
2 spcv.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32spcgv 2774 . 2  |-  ( A  e.  _V  ->  ( A. x ph  ->  ps ) )
41, 3ax-mp 5 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   A.wal 1330    = wceq 1332    e. wcel 1481   _Vcvv 2687
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 2689
This theorem is referenced by:  morex  2869  exmidexmid  4124  exmidsssn  4129  exmidel  4132  rext  4141  ontr2exmid  4444  regexmidlem1  4452  reg2exmid  4455  relop  4693  disjxp1  6137  rdgtfr  6275  ssfiexmid  6774  domfiexmid  6776  diffitest  6785  findcard  6786  fiintim  6821  fisseneq  6824  finomni  7016  exmidomni  7018  exmidlpo  7019  exmidunben  11966  bj-d0clsepcl  13277  bj-inf2vnlem1  13322  subctctexmid  13352
  Copyright terms: Public domain W3C validator