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Theorem comi12 707
Description: Commutation theorem for ->1 and ->2.
Assertion
Ref Expression
comi12 (a ->1 b) C (c ->2 a)
Proof of Theorem comi12
StepHypRef Expression
1 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
2 lea 160 . . . . . . . 8 (a' ^ (c' ^ a')') =< a'
3 leo 158 . . . . . . . 8 a' =< (a' v (a ^ b))
42, 3letr 137 . . . . . . 7 (a' ^ (c' ^ a')') =< (a' v (a ^ b))
54lecom 180 . . . . . 6 (a' ^ (c' ^ a')') C (a' v (a ^ b))
65comcom 453 . . . . 5 (a' v (a ^ b)) C (a' ^ (c' ^ a')')
7 anor3 90 . . . . 5 (a' ^ (c' ^ a')') = (a v (c' ^ a'))'
86, 7cbtr 182 . . . 4 (a' v (a ^ b)) C (a v (c' ^ a'))'
98comcom7 460 . . 3 (a' v (a ^ b)) C (a v (c' ^ a'))
10 df-i2 45 . . . 4 (c ->2 a) = (a v (c' ^ a'))
1110ax-r1 35 . . 3 (a v (c' ^ a')) = (c ->2 a)
129, 11cbtr 182 . 2 (a' v (a ^ b)) C (c ->2 a)
131, 12bctr 181 1 (a ->1 b) C (c ->2 a)
Colors of variables: term
Syntax hints:   C wc 3  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->2 wi2 13
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  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  orbi  842
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