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Theorem notval 145
Description: Value of negation. (Contributed by Mario Carneiro, 8-Oct-2014.)
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 138 . . 3 |- ~ :(* -> *)
2 imval.1 . . 3 |- A:*
31, 2wc 50 . 2 |- (~ A):*
4 df-not 130 . . 3 |- T. |= [~ = \p:* [p:* ==> F.]]
51, 2, 4ceq1 89 . 2 |- T. |= [(~ A) = (\p:* [p:* ==> F.]A)]
6 wim 137 . . . 4 |- ==> :(* -> (* -> *))
7 wv 64 . . . 4 |- p:*:*
8 wfal 135 . . . 4 |- F.:*
96, 7, 8wov 72 . . 3 |- [p:* ==> F.]:*
107, 2weqi 76 . . . . 5 |- [p:* = A]:*
1110id 25 . . . 4 |- [p:* = A] |= [p:* = A]
126, 7, 8, 11oveq1 99 . . 3 |- [p:* = A] |= [[p:* ==> F.] = [A ==> F.]]
139, 2, 12cl 116 . 2 |- T. |= [(\p:* [p:* ==> F.]A) = [A ==> F.]]
143, 5, 13eqtri 95 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 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-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  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:  notval2  159  notnot1  160  con2d  161  alnex  186  exmid  199  notnot  200  ax3  205
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