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Theorem notnot1 150
Description: One side of notnot 187.
Hypothesis
Ref Expression
notval2.1 |- A:*
Assertion
Ref Expression
notnot1 |- A |= (~ (~ A))
Proof of Theorem notnot1
StepHypRef Expression
1 wfal 125 . . . 4 |- F.:*
2 notval2.1 . . . . 5 |- A:*
3 wnot 128 . . . . . 6 |- ~ :(* -> *)
43, 2wc 45 . . . . 5 |- (~ A):*
52, 4simpl 22 . . . 4 |- (A, (~ A)) |= A
62, 4simpr 23 . . . . 5 |- (A, (~ A)) |= (~ A)
75ax-cb1 29 . . . . . 6 |- (A, (~ A)):*
82notval 135 . . . . . 6 |- T. |= [(~ A) = [A ==> F.]]
97, 8a1i 28 . . . . 5 |- (A, (~ A)) |= [(~ A) = [A ==> F.]]
106, 9mpbi 72 . . . 4 |- (A, (~ A)) |= [A ==> F.]
111, 5, 10mpd 146 . . 3 |- (A, (~ A)) |= F.
1211ex 148 . 2 |- A |= [(~ A) ==> F.]
134notval 135 . . 3 |- T. |= [(~ (~ A)) = [(~ A) ==> F.]]
142, 13a1i 28 . 2 |- A |= [(~ (~ A)) = [(~ A) ==> F.]]
1512, 14mpbir 77 1 |- A |= (~ (~ A))
Colors of variables: type var term
Syntax hints:  *hb 3  kc 5   = ke 7  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12  F.tfal 108  ~ tne 110   ==> tim 111
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
This theorem depends on definitions:  df-ov 65  df-al 116  df-fal 117  df-an 118  df-im 119  df-not 120
This theorem is referenced by:  con3d  152  exnal1  175  notnot  187  ax9  199
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