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Theorem u1lemc6 706
Description: Commutation theorem for Sasaki implication.
Assertion
Ref Expression
u1lemc6 (a ->1 b) C (a' ->1 b)
Proof of Theorem u1lemc6
StepHypRef Expression
1 lea 160 . . . . . 6 (a ^ (a' v b')) =< a
2 ax-a1 30 . . . . . 6 a = a''
31, 2lbtr 139 . . . . 5 (a ^ (a' v b')) =< a''
4 leo 158 . . . . 5 a'' =< (a'' v (a' ^ b))
53, 4letr 137 . . . 4 (a ^ (a' v b')) =< (a'' v (a' ^ b))
6 ud1lem0c 277 . . . 4 (a ->1 b)' = (a ^ (a' v b'))
7 df-i1 44 . . . 4 (a' ->1 b) = (a'' v (a' ^ b))
85, 6, 7le3tr1 140 . . 3 (a ->1 b)' =< (a' ->1 b)
98lecom 180 . 2 (a ->1 b)' C (a' ->1 b)
109comcom6 459 1 (a ->1 b) C (a' ->1 b)
Colors of variables: term
Syntax hints:   C wc 3  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12
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-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  negantlem2  849
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