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Theorem notval2 149
Description: Another way two write ~ A, the negation of A.
Hypothesis
Ref Expression
notval2.1 |- A:*
Assertion
Ref Expression
notval2 |- T. |= [(~ A) = [A = F.]]
Proof of Theorem notval2
StepHypRef Expression
1 wnot 128 . . 3 |- ~ :(* -> *)
2 notval2.1 . . 3 |- A:*
31, 2wc 45 . 2 |- (~ A):*
42notval 135 . 2 |- T. |= [(~ A) = [A ==> F.]]
5 wfal 125 . . . . 5 |- F.:*
6 wim 127 . . . . . . 7 |- ==> :(* -> (* -> *))
76, 2, 5wov 64 . . . . . 6 |- [A ==> F.]:*
87, 2simpr 23 . . . . 5 |- ([A ==> F.], A) |= A
97, 2simpl 22 . . . . 5 |- ([A ==> F.], A) |= [A ==> F.]
105, 8, 9mpd 146 . . . 4 |- ([A ==> F.], A) |= F.
112pm2.21 143 . . . . 5 |- F. |= A
1211, 7adantl 51 . . . 4 |- ([A ==> F.], F.) |= A
1310, 12ded 74 . . 3 |- [A ==> F.] |= [A = F.]
1413ax-cb2 30 . . . . . 6 |- [A = F.]:*
1514, 2simpr 23 . . . . 5 |- ([A = F.], A) |= A
1614, 2simpl 22 . . . . 5 |- ([A = F.], A) |= [A = F.]
1715, 16mpbi 72 . . . 4 |- ([A = F.], A) |= F.
1817ex 148 . . 3 |- [A = F.] |= [A ==> F.]
1913, 18dedi 75 . 2 |- T. |= [[A ==> F.] = [A = F.]]
203, 4, 19eqtri 85 1 |- T. |= [(~ A) = [A = F.]]
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
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: (None)
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