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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:(βγ)
distrc.2 A:β
distrc.3 B:α
Assertion
Ref Expression
distrc ⊤⊧[(λx:α (FA)B) = ((λx:α FB)(λx:α AB))]

Proof of Theorem distrc
StepHypRef Expression
1 weq 41 . 2 = :(γ → (γ → ∗))
2 distrc.1 . . . . 5 F:(βγ)
3 distrc.2 . . . . 5 A:β
42, 3wc 50 . . . 4 (FA):γ
54wl 66 . . 3 λx:α (FA):(αγ)
6 distrc.3 . . 3 B:α
75, 6wc 50 . 2 (λx:α (FA)B):γ
82wl 66 . . . 4 λx:α F:(α → (βγ))
98, 6wc 50 . . 3 (λx:α FB):(βγ)
103wl 66 . . . 4 λx:α A:(αβ)
1110, 6wc 50 . . 3 (λx:α AB):β
129, 11wc 50 . 2 ((λx:α FB)(λx:α AB)):γ
133, 6, 2ax-distrc 68 . 2 ⊤⊧(( = (λx:α (FA)B))((λx:α FB)(λx:α AB)))
141, 7, 12, 13dfov2 75 1 ⊤⊧[(λx:α (FA)B) = ((λx:α FB)(λx:α AB))]
 Colors of variables: type var term 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-distrc 68  ax-wov 71 This theorem depends on definitions:  df-ov 73 This theorem is referenced by:  hbc  110
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