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Theorem isfree 176
Description: Derive the metamath "is free" predicate in terms of the HOL "is free" predicate.
Hypotheses
Ref Expression
alnex1.1 |- A:*
isfree.2 |- T. |= [(\x:al Ay:al) = A]
Assertion
Ref Expression
isfree |- T. |= [A ==> (A.\x:al A)]
Distinct variable groups:   y,A   x,y,al

Proof of Theorem isfree
StepHypRef Expression
1 alnex1.1 . . . . 5 |- A:*
21id 25 . . . 4 |- A |= A
3 isfree.2 . . . 4 |- T. |= [(\x:al Ay:al) = A]
42, 3alrimi 170 . . 3 |- A |= (A.\x:al A)
53ax-cb1 29 . . 3 |- T.:*
64, 5adantl 51 . 2 |- (T., A) |= (A.\x:al A)
76ex 148 1 |- T. |= [A ==> (A.\x:al A)]
Colors of variables: type var term
Syntax hints:  tv 1  *hb 3  kc 5  \kl 6   = ke 7  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12   ==> tim 111  A.tal 112
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-refl 39  ax-eqmp 42  ax-ded 43  ax-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  ax-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103  ax-eta 165
This theorem depends on definitions:  df-ov 65  df-al 116  df-an 118  df-im 119
This theorem is referenced by:  ax6  195  ax12  202  ax17  205
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