HOLE Home Higher-Order Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HOLE Home  >  Th. List  >  ax6 Unicode version
Theorem ax6 195
Description: Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113.
Hypothesis
Ref Expression
ax6.1 |- R:*
Assertion
Ref Expression
ax6 |- T. |= [(~ (A.\x:al R)) ==> (A.\x:al (~ (A.\x:al R)))]
Distinct variable group:   al,x
Proof of Theorem ax6
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 wnot 128 . . 3 |- ~ :(* -> *)
2 wal 124 . . . 4 |- A.:((al -> *) -> *)
3 ax6.1 . . . . 5 |- R:*
43wl 59 . . . 4 |- \x:al R:(al -> *)
52, 4wc 45 . . 3 |- (A.\x:al R):*
61, 5wc 45 . 2 |- (~ (A.\x:al R)):*
7 wv 58 . . 3 |- y:al:al
81, 7ax-17 95 . . 3 |- T. |= [(\x:al ~ y:al) = ~ ]
92, 7ax-17 95 . . . 4 |- T. |= [(\x:al A.y:al) = A.]
103, 7ax-hbl1 93 . . . 4 |- T. |= [(\x:al \x:al Ry:al) = \x:al R]
112, 4, 7, 9, 10hbc 100 . . 3 |- T. |= [(\x:al (A.\x:al R)y:al) = (A.\x:al R)]
121, 5, 7, 8, 11hbc 100 . 2 |- T. |= [(\x:al (~ (A.\x:al R))y:al) = (~ (A.\x:al R))]
136, 12isfree 176 1 |- T. |= [(~ (A.\x:al R)) ==> (A.\x:al (~ (A.\x:al R)))]
Colors of variables: type var term
Syntax hints:  tv 1   -> ht 2  *hb 3  kc 5  \kl 6  T.kt 8  [kbr 9   |= wffMMJ2 11  wffMMJ2t 12  ~ tne 110   ==> tim 111  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-distrl 63  ax-hbl1 93  ax-17 95  ax-inst 103  ax-eta 165
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)
  Copyright terms: Public domain W3C validator