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Theorem sklem 230
Description: Soundness lemma.
Hypothesis
Ref Expression
sklem.1 a =< b
Assertion
Ref Expression
sklem (a' v b) = 1
Proof of Theorem sklem
StepHypRef Expression
1 or12 80 . . 3 (a' v (a v b)) = (a v (a' v b))
2 df-t 41 . . . . . 6 1 = (a v a')
32ax-r5 38 . . . . 5 (1 v b) = ((a v a') v b)
43ax-r1 35 . . . 4 ((a v a') v b) = (1 v b)
5 ax-a3 32 . . . 4 ((a v a') v b) = (a v (a' v b))
6 ax-a2 31 . . . 4 (1 v b) = (b v 1)
74, 5, 63tr2 64 . . 3 (a v (a' v b)) = (b v 1)
81, 7ax-r2 36 . 2 (a' v (a v b)) = (b v 1)
9 sklem.1 . . . 4 a =< b
109df-le2 131 . . 3 (a v b) = b
1110lor 70 . 2 (a' v (a v b)) = (a' v b)
12 or1 104 . 2 (b v 1) = 1
138, 11, 123tr2 64 1 (a' v b) = 1
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6  1wt 8
This theorem was proved from axioms:  ax-a2 31  ax-a3 32  ax-a4 33  ax-r1 35  ax-r2 36  ax-r5 38
This theorem depends on definitions:  df-t 41  df-le2 131
This theorem is referenced by:  ska13  241  ska15  244  lei3  246  oaidlem1  294  id5id0  352  u1lemle1  710  u2lemle1  711  u3lemle1  712  u4lemle1  713  u5lemle1  714  lem3.3.3  1052  lem3.3.7i4e1  1069  lem3.3.7i4e2  1070  lem4.6.7  1101
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