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Theorem i3orlem3 554
Description: Lemma for Kalmbach implication OR builder.
Assertion
Ref Expression
i3orlem3 c =< ((a v c) ->3 (b v c))
Proof of Theorem i3orlem3
StepHypRef Expression
1 ax-a2 31 . . . . . 6 ((a v c)' v c) = (c v (a v c)')
21lan 77 . . . . 5 (c ^ ((a v c)' v c)) = (c ^ (c v (a v c)'))
3 anabs 121 . . . . 5 (c ^ (c v (a v c)')) = c
42, 3ax-r2 36 . . . 4 (c ^ ((a v c)' v c)) = c
54ax-r1 35 . . 3 c = (c ^ ((a v c)' v c))
6 leor 159 . . . 4 c =< (a v c)
7 leor 159 . . . . 5 c =< (b v c)
87lelor 166 . . . 4 ((a v c)' v c) =< ((a v c)' v (b v c))
96, 8le2an 169 . . 3 (c ^ ((a v c)' v c)) =< ((a v c) ^ ((a v c)' v (b v c)))
105, 9bltr 138 . 2 c =< ((a v c) ^ ((a v c)' v (b v c)))
11 i3orlem1 552 . 2 ((a v c) ^ ((a v c)' v (b v c))) =< ((a v c) ->3 (b v c))
1210, 11letr 137 1 c =< ((a v c) ->3 (b v c))
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->3 wi3 14
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-i3 46  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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