Theorem ovl 117

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

Hypotheses

Ref	Expression
ovl.1	|- A:ga
ovl.2	|- S:al
ovl.3	|- T:be
ovl.4	|- [x:al = S] |= [A = B]
ovl.5	|- [y:be = T] |= [B = C]

Assertion

Ref	Expression
ovl	|- T. |= [[S\x:al \y:be AT] = C]

Distinct variable groups:   x,B   y,C   x,y,S   y,T   al,x   be,y

Proof of Theorem ovl

Step	Hyp	Ref	Expression
1		ovl.1	. . . . 5 |- A:ga
2	1	wl 66	. . . 4 |- \y:be A:(be -> ga)
3	2	wl 66	. . 3 |- \x:al \y:be A:(al -> (be -> ga))
4		ovl.2	. . 3 |- S:al
5		ovl.3	. . 3 |- T:be
6	3, 4, 5	wov 72	. 2 |- [S\x:al \y:be AT]:ga
7		weq 41	. . . 4 |- = :(ga -> (ga -> *))
8	3, 4	wc 50	. . . . 5 |- (\x:al \y:be AS):(be -> ga)
9	8, 5	wc 50	. . . 4 |- ((\x:al \y:be AS)T):ga
10		wtru 43	. . . . 5 |- T.:*
11	3, 4, 5	df-ov 73	. . . . 5 |- T. |= (( = [S\x:al \y:be AT])((\x:al \y:be AS)T))
12	10, 11	a1i 28	. . . 4 |- T. |= (( = [S\x:al \y:be AT])((\x:al \y:be AS)T))
13	7, 6, 9, 12	dfov2 75	. . 3 |- T. |= [[S\x:al \y:be AT] = ((\x:al \y:be AS)T)]
14	1, 4	distrl 94	. . . . 5 |- T. |= [(\x:al \y:be AS) = \y:be (\x:al AS)]
15	10, 14	a1i 28	. . . 4 |- T. |= [(\x:al \y:be AS) = \y:be (\x:al AS)]
16	8, 5, 15	ceq1 89	. . 3 |- T. |= [((\x:al \y:be AS)T) = (\y:be (\x:al AS)T)]
17	6, 13, 16	eqtri 95	. 2 |- T. |= [[S\x:al \y:be AT] = (\y:be (\x:al AS)T)]
18	1	wl 66	. . . 4 |- \x:al A:(al -> ga)
19	18, 4	wc 50	. . 3 |- (\x:al AS):ga
20		wv 64	. . . . . 6 |- y:be:be
21	20, 5	weqi 76	. . . . 5 |- [y:be = T]:*
22		ovl.4	. . . . . 6 |- [x:al = S] |= [A = B]
23	1, 4, 22	cl 116	. . . . 5 |- T. |= [(\x:al AS) = B]
24	21, 23	a1i 28	. . . 4 |- [y:be = T] |= [(\x:al AS) = B]
25		ovl.5	. . . 4 |- [y:be = T] |= [B = C]
26	19, 24, 25	eqtri 95	. . 3 |- [y:be = T] |= [(\x:al AS) = C]
27	19, 5, 26	cl 116	. 2 |- T. |= [(\y:be (\x:al AS)T) = C]
28	6, 17, 27	eqtri 95	1 |- T. |= [[S\x:al \y:be AT] = C]

Syntax hints:  tv 1   -> ht 2  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-distrl 70  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:  imval  146  orval  147  anval  148  dfan2  154