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Theorem oal42 935
Description: Derivation of Godowski/Greechie Eq. II from Eq. IV.
Hypothesis
Ref Expression
oal42.1 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< ((b' ^ (a ->2 b)) v (c' ^ (a ->2 c)))
Assertion
Ref Expression
oal42 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< a'
Proof of Theorem oal42
StepHypRef Expression
1 oal42.1 . . 3 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< ((b' ^ (a ->2 b)) v (c' ^ (a ->2 c)))
2 ancom 74 . . . . 5 (b' ^ (a ->2 b)) = ((a ->2 b) ^ b')
3 u2lemanb 616 . . . . 5 ((a ->2 b) ^ b') = (a' ^ b')
42, 3ax-r2 36 . . . 4 (b' ^ (a ->2 b)) = (a' ^ b')
5 ancom 74 . . . . 5 (c' ^ (a ->2 c)) = ((a ->2 c) ^ c')
6 u2lemanb 616 . . . . 5 ((a ->2 c) ^ c') = (a' ^ c')
75, 6ax-r2 36 . . . 4 (c' ^ (a ->2 c)) = (a' ^ c')
84, 72or 72 . . 3 ((b' ^ (a ->2 b)) v (c' ^ (a ->2 c))) = ((a' ^ b') v (a' ^ c'))
91, 8lbtr 139 . 2 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< ((a' ^ b') v (a' ^ c'))
10 lea 160 . . 3 (a' ^ b') =< a'
11 lea 160 . . 3 (a' ^ c') =< a'
1210, 11lel2or 170 . 2 ((a' ^ b') v (a' ^ c')) =< a'
139, 12letr 137 1 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< a'
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->2 wi2 13
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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  oa43v  1028  oa63v  1032
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