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.

Step	Hyp	Ref	Expression
1		cl.1	. 2 |- A:be
2		cl.2	. 2 |- C:al
3		cl.3	. 2 |- [x:al = C] |= [A = B]
4	1, 3	eqtypi 69	. . 3 |- B:be
5		wv 58	. . 3 |- y:al:al
6	4, 5	ax-17 95	. 2 |- T. |= [(\x:al By:al) = B]
7	2, 5	ax-17 95	. 2 |- T. |= [(\x:al Cy:al) = C]
8	1, 2, 3, 6, 7	clf 105	1 |- T. |= [(\x:al AC) = B]

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