HOLE Home Higher-Order Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HOLE Home  >  Th. List  >  cl Unicode version
Theorem cl 106
Description: Evaluate a lambda expression.
Hypotheses
Ref Expression
cl.1 |- A:be
cl.2 |- C:al
cl.3 |- [x:al = C] |= [A = B]
Assertion
Ref Expression
cl |- T. |= [(\x:al AC) = B]
Distinct variable groups:   x,B   x,C   al,x
Proof of Theorem cl
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 cl.1 . 2 |- A:be
2 cl.2 . 2 |- C:al
3 cl.3 . 2 |- [x:al = C] |= [A = B]
41, 3eqtypi 69 . . 3 |- B:be
5 wv 58 . . 3 |- y:al:al
64, 5ax-17 95 . 2 |- T. |= [(\x:al By:al) = B]
72, 5ax-17 95 . 2 |- T. |= [(\x:al Cy:al) = C]
81, 2, 3, 6, 7clf 105 1 |- T. |= [(\x:al AC) = B]
Colors of variables: type var term
Syntax hints:  tv 1  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12
This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-refl 39  ax-eqmp 42  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-hbl1 93  ax-17 95  ax-inst 103
This theorem depends on definitions:  df-ov 65
This theorem is referenced by:  ovl  107  alval  132  exval  133  euval  134  notval  135  cla4v  142  dfan2  144  cla4ev  159  exmid  186  axpow  208
  Copyright terms: Public domain W3C validator