Theorem clf 115

Description: Evaluate a lambda expression. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypotheses

Ref	Expression
clf.1	|- A:be
clf.2	|- C:al
clf.3	|- [x:al = C] |= [A = B]
clf.4	|- T. |= [(\x:al By:al) = B]
clf.5	|- T. |= [(\x:al Cy:al) = C]

Assertion

Ref	Expression
clf	|- T. |= [(\x:al AC) = B]

Distinct variable groups:   y,A   y,B   y,C   x,y,al

Proof of Theorem clf

Step	Hyp	Ref	Expression
1		clf.2	. 2 |- C:al
2		clf.1	. . . . 5 |- A:be
3	2	wl 66	. . . 4 |- \x:al A:(al -> be)
4	3, 1	wc 50	. . 3 |- (\x:al AC):be
5		clf.3	. . . 4 |- [x:al = C] |= [A = B]
6	2, 5	eqtypi 78	. . 3 |- B:be
7	4, 6	weqi 76	. 2 |- [(\x:al AC) = B]:*
8		clf.4	. . . 4 |- T. |= [(\x:al By:al) = B]
9	8	ax-cb1 29	. . 3 |- T.:*
10	2	beta 92	. . 3 |- T. |= [(\x:al Ax:al) = A]
11	9, 10	a1i 28	. 2 |- T. |= [(\x:al Ax:al) = A]
12		weq 41	. . 3 |- = :(be -> (be -> *))
13		wv 64	. . 3 |- y:al:al
14	12, 13, 9	a17i 106	. . 3 |- T. |= [(\x:al = y:al) = = ]
15	2, 13, 9	hbl1 104	. . . 4 |- T. |= [(\x:al \x:al Ay:al) = \x:al A]
16		clf.5	. . . 4 |- T. |= [(\x:al Cy:al) = C]
17	3, 1, 13, 15, 16	hbc 110	. . 3 |- T. |= [(\x:al (\x:al AC)y:al) = (\x:al AC)]
18	12, 4, 13, 6, 14, 17, 8	hbov 111	. 2 |- T. |= [(\x:al [(\x:al AC) = B]y:al) = [(\x:al AC) = B]]
19		wv 64	. . . 4 |- x:al:al
20	3, 19	wc 50	. . 3 |- (\x:al Ax:al):be
21	19, 1	weqi 76	. . . . 5 |- [x:al = C]:*
22	21	id 25	. . . 4 |- [x:al = C] |= [x:al = C]
23	3, 19, 22	ceq2 90	. . 3 |- [x:al = C] |= [(\x:al Ax:al) = (\x:al AC)]
24	12, 20, 2, 23, 5	oveq12 100	. 2 |- [x:al = C] |= [[(\x:al Ax:al) = A] = [(\x:al AC) = B]]
25	1, 7, 11, 18, 24	insti 114	1 |- T. |= [(\x:al AC) = B]

Syntax hints:  tv 1   -> ht 2  *hb 3  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-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113

This theorem depends on definitions:  df-ov 73

This theorem is referenced by:  cl  116  cbvf  179  exmid  199  axrep  220