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Theorem isfree 188
 Description: Derive the metamath "is free" predicate in terms of the HOL "is free" predicate. (Contributed by Mario Carneiro, 9-Oct-2014.)
Hypotheses
Ref Expression
alnex1.1 A:∗
isfree.2 ⊤⊧[(λx:α Ay:α) = A]
Assertion
Ref Expression
isfree ⊤⊧[A ⇒ (λx:α A)]
Distinct variable groups:   y,A   x,y,α

Proof of Theorem isfree
StepHypRef Expression
1 alnex1.1 . . . . 5 A:∗
21id 25 . . . 4 AA
3 isfree.2 . . . 4 ⊤⊧[(λx:α Ay:α) = A]
42, 3alrimi 182 . . 3 A⊧(λx:α A)
53ax-cb1 29 . . 3 ⊤:∗
64, 5adantl 56 . 2 (⊤, A)⊧(λx:α A)
76ex 158 1 ⊤⊧[A ⇒ (λx:α A)]
 Colors of variables: type var term Syntax hints:  tv 1  ∗hb 3  kc 5  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12   ⇒ tim 121  ∀tal 122 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  df-an 128  df-im 129 This theorem is referenced by:  ax6  208  ax12  215  ax17m  218
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