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Theorem i3orlem5 556
Description: Lemma for Kalmbach implication OR builder.
Assertion
Ref Expression
i3orlem5 ((a' ^ b') ^ c') =< ((a v c) ->3 (b v c))
Proof of Theorem i3orlem5
StepHypRef Expression
1 leo 158 . 2 ((a v c)' ^ (b v c)') =< (((a v c)' ^ (b v c)') v (((a v c)' v (b v c)) ^ ((a v c) v ((a v c)' ^ (b v c)))))
2 anandir 115 . . 3 ((a' ^ b') ^ c') = ((a' ^ c') ^ (b' ^ c'))
3 oran 87 . . . . . 6 (a v c) = (a' ^ c')'
43con2 67 . . . . 5 (a v c)' = (a' ^ c')
54ax-r1 35 . . . 4 (a' ^ c') = (a v c)'
6 oran 87 . . . . . 6 (b v c) = (b' ^ c')'
76con2 67 . . . . 5 (b v c)' = (b' ^ c')
87ax-r1 35 . . . 4 (b' ^ c') = (b v c)'
95, 82an 79 . . 3 ((a' ^ c') ^ (b' ^ c')) = ((a v c)' ^ (b v c)')
102, 9ax-r2 36 . 2 ((a' ^ b') ^ c') = ((a v c)' ^ (b v c)')
11 df2i3 498 . 2 ((a v c) ->3 (b v c)) = (((a v c)' ^ (b v c)') v (((a v c)' v (b v c)) ^ ((a v c) v ((a v c)' ^ (b v c)))))
121, 10, 11le3tr1 140 1 ((a' ^ b') ^ 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-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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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