Theorem distrl 94

Description: Distribution of lambda abstraction over substitution. (Contributed by Mario Carneiro, 8-Oct-2014.)

Hypotheses

Ref	Expression
distrl.1	⊢ A:γ
distrl.2	⊢ B:α

Assertion

Ref	Expression
distrl	⊢ ⊤⊧[(λx:α λy:β AB) = λy:β (λx:α AB)]

Distinct variable groups:   x,y   y,B

Proof of Theorem distrl

Step	Hyp	Ref	Expression
1		weq 41	. 2 ⊢ = :((β → γ) → ((β → γ) → ∗))
2		distrl.1	. . . . 5 ⊢ A:γ
3	2	wl 66	. . . 4 ⊢ λy:β A:(β → γ)
4	3	wl 66	. . 3 ⊢ λx:α λy:β A:(α → (β → γ))
5		distrl.2	. . 3 ⊢ B:α
6	4, 5	wc 50	. 2 ⊢ (λx:α λy:β AB):(β → γ)
7	2	wl 66	. . . 4 ⊢ λx:α A:(α → γ)
8	7, 5	wc 50	. . 3 ⊢ (λx:α AB):γ
9	8	wl 66	. 2 ⊢ λy:β (λx:α AB):(β → γ)
10	2, 5	ax-distrl 70	. 2 ⊢ ⊤⊧(( = (λx:α λy:β AB))λy:β (λx:α AB))
11	1, 6, 9, 10	dfov2 75	1 ⊢ ⊤⊧[(λx:α λy:β AB) = λy:β (λx:α AB)]

Syntax hints:   → ht 2  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-trud 26  ax-cb1 29  ax-cb2 30  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wl 65  ax-distrl 70  ax-wov 71

This theorem depends on definitions:  df-ov 73

This theorem is referenced by:  hbl  112  ovl  117