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Theorem i1or 345
Description: Lemma for disjunction of ->1.
Assertion
Ref Expression
i1or ((c ->1 a) v (c ->1 b)) =< (c ->1 (a v b))
Proof of Theorem i1or
StepHypRef Expression
1 df-i1 44 . . . 4 (c ->1 a) = (c' v (c ^ a))
2 leo 158 . . . . . 6 a =< (a v b)
32lelan 167 . . . . 5 (c ^ a) =< (c ^ (a v b))
43lelor 166 . . . 4 (c' v (c ^ a)) =< (c' v (c ^ (a v b)))
51, 4bltr 138 . . 3 (c ->1 a) =< (c' v (c ^ (a v b)))
6 df-i1 44 . . . 4 (c ->1 b) = (c' v (c ^ b))
7 leor 159 . . . . . 6 b =< (a v b)
87lelan 167 . . . . 5 (c ^ b) =< (c ^ (a v b))
98lelor 166 . . . 4 (c' v (c ^ b)) =< (c' v (c ^ (a v b)))
106, 9bltr 138 . . 3 (c ->1 b) =< (c' v (c ^ (a v b)))
115, 10lel2or 170 . 2 ((c ->1 a) v (c ->1 b)) =< (c' v (c ^ (a v b)))
12 df-i1 44 . . 3 (c ->1 (a v b)) = (c' v (c ^ (a v b)))
1312ax-r1 35 . 2 (c' v (c ^ (a v b))) = (c ->1 (a v b))
1411, 13lbtr 139 1 ((c ->1 a) v (c ->1 b)) =< (c ->1 (a v b))
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131
This theorem is referenced by:  orbile  843
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