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Theorem mhlemlem2 875
Description: Lemma for Lemma 7.1 of Kalmbach, p. 91.
Hypothesis
Ref Expression
mhlem.1 (a v b) =< (c v d)'
Assertion
Ref Expression
mhlemlem2 (((a v b) v d) ^ (b v (c v d))) = (b v d)
Proof of Theorem mhlemlem2
StepHypRef Expression
1 ax-a2 31 . . . 4 (a v b) = (b v a)
21ax-r5 38 . . 3 ((a v b) v d) = ((b v a) v d)
3 ax-a2 31 . . . 4 (c v d) = (d v c)
43lor 70 . . 3 (b v (c v d)) = (b v (d v c))
52, 42an 79 . 2 (((a v b) v d) ^ (b v (c v d))) = (((b v a) v d) ^ (b v (d v c)))
6 mhlem.1 . . . 4 (a v b) =< (c v d)'
7 ax-a2 31 . . . 4 (b v a) = (a v b)
8 ax-a2 31 . . . . 5 (d v c) = (c v d)
98ax-r4 37 . . . 4 (d v c)' = (c v d)'
106, 7, 9le3tr1 140 . . 3 (b v a) =< (d v c)'
1110mhlemlem1 874 . 2 (((b v a) v d) ^ (b v (d v c))) = (b v d)
125, 11ax-r2 36 1 (((a v b) v d) ^ (b v (c v d))) = (b v d)
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7
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-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  mhlem  876
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