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Theorem notnot1 160
Description: One side of notnot 200. (Contributed by Mario Carneiro, 10-Oct-2014.)
Hypothesis
Ref Expression
notval2.1 |- A:*
Assertion
Ref Expression
notnot1 |- A |= (~ (~ A))

Proof of Theorem notnot1
StepHypRef Expression
1 wfal 135 . . . 4 |- F.:*
2 notval2.1 . . . . 5 |- A:*
3 wnot 138 . . . . . 6 |- ~ :(* -> *)
43, 2wc 50 . . . . 5 |- (~ A):*
52, 4simpl 22 . . . 4 |- (A, (~ A)) |= A
62, 4simpr 23 . . . . 5 |- (A, (~ A)) |= (~ A)
75ax-cb1 29 . . . . . 6 |- (A, (~ A)):*
82notval 145 . . . . . 6 |- T. |= [(~ A) = [A ==> F.]]
97, 8a1i 28 . . . . 5 |- (A, (~ A)) |= [(~ A) = [A ==> F.]]
106, 9mpbi 82 . . . 4 |- (A, (~ A)) |= [A ==> F.]
111, 5, 10mpd 156 . . 3 |- (A, (~ A)) |= F.
1211ex 158 . 2 |- A |= [(~ A) ==> F.]
134notval 145 . . 3 |- T. |= [(~ (~ A)) = [(~ A) ==> F.]]
142, 13a1i 28 . 2 |- A |= [(~ (~ A)) = [(~ A) ==> F.]]
1512, 14mpbir 87 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 118  ~ tne 120   ==> tim 121
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
This theorem depends on definitions:  df-ov 73  df-al 126  df-fal 127  df-an 128  df-im 129  df-not 130
This theorem is referenced by:  con3d  162  exnal1  187  notnot  200  ax9  212
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