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Theorem notval 135
Description: Value of negation.
Hypothesis
Ref Expression
imval.1 |- A:*
Assertion
Ref Expression
notval |- T. |= [(~ A) = [A ==> F.]]
Proof of Theorem notval
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 wnot 128 . . 3 |- ~ :(* -> *)
2 imval.1 . . 3 |- A:*
31, 2wc 45 . 2 |- (~ A):*
4 df-not 120 . . 3 |- T. |= [~ = \p:* [p:* ==> F.]]
51, 2, 4ceq1 79 . 2 |- T. |= [(~ A) = (\p:* [p:* ==> F.]A)]
6 wim 127 . . . 4 |- ==> :(* -> (* -> *))
7 wv 58 . . . 4 |- p:*:*
8 wfal 125 . . . 4 |- F.:*
96, 7, 8wov 64 . . 3 |- [p:* ==> F.]:*
107, 2weqi 68 . . . . 5 |- [p:* = A]:*
1110id 25 . . . 4 |- [p:* = A] |= [p:* = A]
126, 7, 8, 11oveq1 89 . . 3 |- [p:* = A] |= [[p:* ==> F.] = [A ==> F.]]
139, 2, 12cl 106 . 2 |- T. |= [(\p:* [p:* ==> F.]A) = [A ==> F.]]
143, 5, 13eqtri 85 1 |- T. |= [(~ A) = [A ==> F.]]
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  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-ceq 46  ax-beta 60  ax-distrc 61  ax-leq 62  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:  notval2  149  notnot1  150  con2d  151  alnex  174  exmid  186  notnot  187  ax3  192
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