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Theorem pm2.21 143
Description: A falsehood implies anything.
Hypothesis
Ref Expression
pm2.21.1 |- A:*
Assertion
Ref Expression
pm2.21 |- F. |= A
Proof of Theorem pm2.21
Dummy variable p is distinct from all other variables.
StepHypRef Expression
1 wfal 125 . . . 4 |- F.:*
21id 25 . . 3 |- F. |= F.
3 df-fal 117 . . . 4 |- T. |= [F. = (A.\p:* p:*)]
41, 3a1i 28 . . 3 |- F. |= [F. = (A.\p:* p:*)]
52, 4mpbi 72 . 2 |- F. |= (A.\p:* p:*)
6 wv 58 . . 3 |- p:*:*
7 pm2.21.1 . . 3 |- A:*
86, 7weqi 68 . . . 4 |- [p:* = A]:*
98id 25 . . 3 |- [p:* = A] |= [p:* = A]
106, 7, 9cla4v 142 . 2 |- (A.\p:* p:*) |= A
115, 10syl 16 1 |- F. |= A
Colors of variables: type var term
Syntax hints:  tv 1  *hb 3  kc 5  \kl 6   = ke 7  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  F.tfal 108  A.tal 112
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-hbl1 93  ax-17 95  ax-inst 103
This theorem depends on definitions:  df-ov 65  df-al 116  df-fal 117
This theorem is referenced by:  notval2  149  notnot  187  ax3  192
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