Theorem clf 115

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

Hypotheses

Ref	Expression
clf.1	⊢ A:β
clf.2	⊢ C:α
clf.3	⊢ [x:α = C]⊧[A = B]
clf.4	⊢ ⊤⊧[(λx:α By:α) = B]
clf.5	⊢ ⊤⊧[(λx:α Cy:α) = C]

Assertion

Ref	Expression
clf	⊢ ⊤⊧[(λx:α AC) = B]

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

Proof of Theorem clf

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

Syntax hints:  tv 1   → ht 2  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤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