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Theorem notnot 187
Description: Rule of double negation.
Hypothesis
Ref Expression
exmid.1 |- A:*
Assertion
Ref Expression
notnot |- T. |= [A = (~ (~ A))]
Proof of Theorem notnot
StepHypRef Expression
1 exmid.1 . . 3 |- A:*
21notnot1 150 . 2 |- A |= (~ (~ A))
3 wnot 128 . . . 4 |- ~ :(* -> *)
43, 1wc 45 . . 3 |- (~ A):*
52ax-cb2 30 . . . 4 |- (~ (~ A)):*
61exmid 186 . . . 4 |- T. |= [A \/ (~ A)]
75, 6a1i 28 . . 3 |- (~ (~ A)) |= [A \/ (~ A)]
85, 1simpr 23 . . 3 |- ((~ (~ A)), A) |= A
9 wfal 125 . . . . 5 |- F.:*
105id 25 . . . . . 6 |- (~ (~ A)) |= (~ (~ A))
114notval 135 . . . . . . 7 |- T. |= [(~ (~ A)) = [(~ A) ==> F.]]
125, 11a1i 28 . . . . . 6 |- (~ (~ A)) |= [(~ (~ A)) = [(~ A) ==> F.]]
1310, 12mpbi 72 . . . . 5 |- (~ (~ A)) |= [(~ A) ==> F.]
144, 9, 13imp 147 . . . 4 |- ((~ (~ A)), (~ A)) |= F.
151pm2.21 143 . . . 4 |- F. |= A
1614, 15syl 16 . . 3 |- ((~ (~ A)), (~ A)) |= A
171, 4, 1, 7, 8, 16ecase 153 . 2 |- (~ (~ A)) |= A
182, 17dedi 75 1 |- T. |= [A = (~ (~ A))]
Colors of variables: type var term
Syntax hints:  *hb 3  kc 5   = ke 7  T.kt 8  [kbr 9  kct 10   |= wffMMJ2 11  wffMMJ2t 12  F.tfal 108  ~ tne 110   ==> tim 111   \/ tor 114
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-ac 183
This theorem depends on definitions:  df-ov 65  df-al 116  df-fal 117  df-an 118  df-im 119  df-not 120  df-or 122
This theorem is referenced by:  exnal  188
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