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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
StepHypRef Expression
1 weq 41 . 2 |- = :(ga -> (ga -> *))
2 distrc.1 . . . . 5 |- F:(be -> ga)
3 distrc.2 . . . . 5 |- A:be
42, 3wc 50 . . . 4 |- (FA):ga
54wl 66 . . 3 |- \x:al (FA):(al -> ga)
6 distrc.3 . . 3 |- B:al
75, 6wc 50 . 2 |- (\x:al (FA)B):ga
82wl 66 . . . 4 |- \x:al F:(al -> (be -> ga))
98, 6wc 50 . . 3 |- (\x:al FB):(be -> ga)
103wl 66 . . . 4 |- \x:al A:(al -> be)
1110, 6wc 50 . . 3 |- (\x:al AB):be
129, 11wc 50 . 2 |- ((\x:al FB)(\x:al AB)):ga
133, 6, 2ax-distrc 68 . 2 |- T. |= (( = (\x:al (FA)B))((\x:al FB)(\x:al AB)))
141, 7, 12, 13dfov2 75 1 |- T. |= [(\x:al (FA)B) = ((\x:al FB)(\x:al AB))]
Colors of variables: type var term
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
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