Theorem distrc 93

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

Hypotheses

Ref	Expression
distrc.1	|- F:(be -> ga)
distrc.2	|- A:be
distrc.3	|- B:al

Assertion

Ref	Expression
distrc	|- T. |= [(\x:al (FA)B) = ((\x:al FB)(\x:al AB))]

Proof of Theorem distrc

Step	Hyp	Ref	Expression
1		weq 41	. 2 |- = :(ga -> (ga -> *))
2		distrc.1	. . . . 5 |- F:(be -> ga)
3		distrc.2	. . . . 5 |- A:be
4	2, 3	wc 50	. . . 4 |- (FA):ga
5	4	wl 66	. . 3 |- \x:al (FA):(al -> ga)
6		distrc.3	. . 3 |- B:al
7	5, 6	wc 50	. 2 |- (\x:al (FA)B):ga
8	2	wl 66	. . . 4 |- \x:al F:(al -> (be -> ga))
9	8, 6	wc 50	. . 3 |- (\x:al FB):(be -> ga)
10	3	wl 66	. . . 4 |- \x:al A:(al -> be)
11	10, 6	wc 50	. . 3 |- (\x:al AB):be
12	9, 11	wc 50	. 2 |- ((\x:al FB)(\x:al AB)):ga
13	3, 6, 2	ax-distrc 68	. 2 |- T. |= (( = (\x:al (FA)B))((\x:al FB)(\x:al AB)))
14	1, 7, 12, 13	dfov2 75	1 |- T. |= [(\x:al (FA)B) = ((\x:al FB)(\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-distrc 68  ax-wov 71

This theorem depends on definitions:  df-ov 73

This theorem is referenced by:  hbc  110