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Theorem hbl 112
 Description: Hypothesis builder for lambda abstraction. (Contributed by Mario Carneiro, 8-Oct-2014.)
Hypotheses
Ref Expression
hbl.1 A:γ
hbl.2 B:α
hbl.3 R⊧[(λx:α AB) = A]
Assertion
Ref Expression
hbl R⊧[(λx:α λy:β AB) = λy:β A]
Distinct variable groups:   x,y   y,B   y,R

Proof of Theorem hbl
StepHypRef Expression
1 hbl.1 . . . . 5 A:γ
21wl 66 . . . 4 λy:β A:(βγ)
32wl 66 . . 3 λx:α λy:β A:(α → (βγ))
4 hbl.2 . . 3 B:α
53, 4wc 50 . 2 (λx:α λy:β AB):(βγ)
6 hbl.3 . . . 4 R⊧[(λx:α AB) = A]
76ax-cb1 29 . . 3 R:∗
81, 4distrl 94 . . 3 ⊤⊧[(λx:α λy:β AB) = λy:β (λx:α AB)]
97, 8a1i 28 . 2 R⊧[(λx:α λy:β AB) = λy:β (λx:α AB)]
101wl 66 . . . 4 λx:α A:(αγ)
1110, 4wc 50 . . 3 (λx:α AB):γ
1211, 6leq 91 . 2 R⊧[λy:β (λx:α AB) = λy:β A]
135, 9, 12eqtri 95 1 R⊧[(λx:α λy:β AB) = λy:β A]
 Colors of variables: type var term Syntax hints:   → ht 2  kc 5  λkl 6   = ke 7  [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-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77 This theorem depends on definitions:  df-ov 73 This theorem is referenced by:  cbvf  179  ax7  209  axrep  220
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