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Theorem i3orlem2 553
Description: Lemma for Kalmbach implication OR builder.
Assertion
Ref Expression
i3orlem2 (a ^ b) =< ((a v c) ->3 (b v c))
Proof of Theorem i3orlem2
StepHypRef Expression
1 leo 158 . . 3 a =< (a v c)
2 leo 158 . . 3 b =< (b v c)
31, 2le2an 169 . 2 (a ^ b) =< ((a v c) ^ (b v c))
4 leor 159 . . . 4 ((a v c) ^ (b v c)) =< (((a v c) ^ (a v c)') v ((a v c) ^ (b v c)))
5 ledi 174 . . . 4 (((a v c) ^ (a v c)') v ((a v c) ^ (b v c))) =< ((a v c) ^ ((a v c)' v (b v c)))
64, 5letr 137 . . 3 ((a v c) ^ (b v c)) =< ((a v c) ^ ((a v c)' v (b v c)))
7 i3orlem1 552 . . 3 ((a v c) ^ ((a v c)' v (b v c))) =< ((a v c) ->3 (b v c))
86, 7letr 137 . 2 ((a v c) ^ (b v c)) =< ((a v c) ->3 (b v c))
93, 8letr 137 1 (a ^ b) =< ((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:  i3orlem6  557
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