QLE Home Quantum Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  QLE Home  >  Th. List  >  neg3ant1 Unicode version
Theorem neg3ant1 866
Description: Lemma for negated antecedent identity.
Hypothesis
Ref Expression
neg3ant.1 (a ->3 c) = (b ->3 c)
Assertion
Ref Expression
neg3ant1 (a ->1 c) = (b ->1 c)
Proof of Theorem neg3ant1
StepHypRef Expression
1 neg3ant.1 . . . . . 6 (a ->3 c) = (b ->3 c)
21neg3antlem2 865 . . . . 5 a' =< (b ->1 c)
31neg3antlem1 864 . . . . 5 (a ^ c) =< (b ->1 c)
42, 3lel2or 170 . . . 4 (a' v (a ^ c)) =< (b ->1 c)
5 df-i1 44 . . . 4 (b ->1 c) = (b' v (b ^ c))
64, 5lbtr 139 . . 3 (a' v (a ^ c)) =< (b' v (b ^ c))
71ax-r1 35 . . . . . 6 (b ->3 c) = (a ->3 c)
87neg3antlem2 865 . . . . 5 b' =< (a ->1 c)
97neg3antlem1 864 . . . . 5 (b ^ c) =< (a ->1 c)
108, 9lel2or 170 . . . 4 (b' v (b ^ c)) =< (a ->1 c)
11 df-i1 44 . . . 4 (a ->1 c) = (a' v (a ^ c))
1210, 11lbtr 139 . . 3 (b' v (b ^ c)) =< (a' v (a ^ c))
136, 12lebi 145 . 2 (a' v (a ^ c)) = (b' v (b ^ c))
1413, 11, 53tr1 63 1 (a ->1 c) = (b ->1 c)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->3 wi3 14
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator