Higher-Order Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HOLE Home  >  Th. List  >  cbv GIF version

Theorem cbv 180
 Description: Change bound variables in a lambda abstraction. (Contributed by Mario Carneiro, 8-Oct-2014.)
Hypotheses
Ref Expression
cbv.1 A:β
cbv.2 [x:α = y:α]⊧[A = B]
Assertion
Ref Expression
cbv ⊤⊧[λx:α A = λy:α B]
Distinct variable groups:   y,A   x,B   x,y,α   β,y

Proof of Theorem cbv
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 cbv.1 . 2 A:β
2 wv 64 . . 3 z:α:α
31, 2ax-17 105 . 2 ⊤⊧[(λy:α Az:α) = A]
4 cbv.2 . . . 4 [x:α = y:α]⊧[A = B]
51, 4eqtypi 78 . . 3 B:β
65, 2ax-17 105 . 2 ⊤⊧[(λx:α Bz:α) = B]
71, 3, 6, 4cbvf 179 1 ⊤⊧[λx:α A = λy:α B]
 Colors of variables: type var term Syntax hints:  tv 1  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-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113  ax-eta 177 This theorem depends on definitions:  df-ov 73  df-al 126 This theorem is referenced by:  ax10  213
 Copyright terms: Public domain W3C validator