Theorem distrl 94

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

Hypotheses

Ref	Expression
distrl.1	|- A:ga
distrl.2	|- B:al

Assertion

Ref	Expression
distrl	|- T. |= [(\x:al \y:be AB) = \y:be (\x:al AB)]

Distinct variable groups:   x,y   y,B

Proof of Theorem distrl

Step	Hyp	Ref	Expression
1		weq 41	. 2 |- = :((be -> ga) -> ((be -> ga) -> *))
2		distrl.1	. . . . 5 |- A:ga
3	2	wl 66	. . . 4 |- \y:be A:(be -> ga)
4	3	wl 66	. . 3 |- \x:al \y:be A:(al -> (be -> ga))
5		distrl.2	. . 3 |- B:al
6	4, 5	wc 50	. 2 |- (\x:al \y:be AB):(be -> ga)
7	2	wl 66	. . . 4 |- \x:al A:(al -> ga)
8	7, 5	wc 50	. . 3 |- (\x:al AB):ga
9	8	wl 66	. 2 |- \y:be (\x:al AB):(be -> ga)
10	2, 5	ax-distrl 70	. 2 |- T. |= (( = (\x:al \y:be AB))\y:be (\x:al AB))
11	1, 6, 9, 10	dfov2 75	1 |- T. |= [(\x:al \y:be AB) = \y:be (\x:al AB)]

Syntax hints:   -> 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-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