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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
StepHypRef Expression
1 weq 41 . 2 = :((βγ) → ((βγ) → ∗))
2 distrl.1 . . . . 5 A:γ
32wl 66 . . . 4 λy:β A:(βγ)
43wl 66 . . 3 λx:α λy:β A:(α → (βγ))
5 distrl.2 . . 3 B:α
64, 5wc 50 . 2 (λx:α λy:β AB):(βγ)
72wl 66 . . . 4 λx:α A:(αγ)
87, 5wc 50 . . 3 (λx:α AB):γ
98wl 66 . 2 λy:β (λx:α AB):(βγ)
102, 5ax-distrl 70 . 2 ⊤⊧(( = (λx:α λy:β AB))λy:β (λx:α AB))
111, 6, 9, 10dfov2 75 1 ⊤⊧[(λx:α λy:β AB) = λy:β (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-distrl 70  ax-wov 71 This theorem depends on definitions:  df-ov 73 This theorem is referenced by:  hbl  112  ovl  117
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