Theorem cl 116

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

Hypotheses

Ref	Expression
cl.1	⊢ A:β
cl.2	⊢ C:α
cl.3	⊢ [x:α = C]⊧[A = B]

Assertion

Ref	Expression
cl	⊢ ⊤⊧[(λx:α AC) = B]

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

Proof of Theorem cl

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cl.1	. 2 ⊢ A:β
2		cl.2	. 2 ⊢ C:α
3		cl.3	. 2 ⊢ [x:α = C]⊧[A = B]
4	1, 3	eqtypi 78	. . 3 ⊢ B:β
5		wv 64	. . 3 ⊢ y:α:α
6	4, 5	ax-17 105	. 2 ⊢ ⊤⊧[(λx:α By:α) = B]
7	2, 5	ax-17 105	. 2 ⊢ ⊤⊧[(λx:α Cy:α) = C]
8	1, 2, 3, 6, 7	clf 115	1 ⊢ ⊤⊧[(λx:α AC) = B]

Syntax hints:  tv 1  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:  ovl  117  alval  142  exval  143  euval  144  notval  145  cla4v  152  dfan2  154  cla4ev  169  exmid  199  axpow  221