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Theorem cl 116
 Description: Evaluate a lambda expression. (Contributed by Mario Carneiro, 8-Oct-2014.)
Hypotheses
Ref Expression
cl.1 A:β
cl.2 C:α
cl.3 [x:α = C]⊧[A = B]
Assertion
Ref Expression
cl ⊤⊧[(λx:α AC) = B]
Distinct variable groups:   x,B   x,C   α,x

Proof of Theorem cl
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 cl.1 . 2 A:β
2 cl.2 . 2 C:α
3 cl.3 . 2 [x:α = C]⊧[A = B]
41, 3eqtypi 78 . . 3 B:β
5 wv 64 . . 3 y:α:α
64, 5ax-17 105 . 2 ⊤⊧[(λx:α By:α) = B]
72, 5ax-17 105 . 2 ⊤⊧[(λx:α Cy:α) = C]
81, 2, 3, 6, 7clf 115 1 ⊤⊧[(λx:α AC) = B]
 Colors of variables: type var term Syntax hints:  tv 1  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-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-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113 This theorem depends on definitions:  df-ov 73 This theorem is referenced by:  ovl  117  alval  142  exval  143  euval  144  notval  145  cla4v  152  dfan2  154  cla4ev  169  exmid  199  axpow  221
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