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Theorem comanb 872
Description: Biconditional commutation law.
Assertion
Ref Expression
comanb (a ^ b) C ((a == c) ^ (b == c))
Proof of Theorem comanb
StepHypRef Expression
1 lea 160 . . . 4 (((a v c)' v ((a ^ b) ^ c)) ^ (b ->1 c)) =< ((a v c)' v ((a ^ b) ^ c))
2 lea 160 . . . . . . 7 (a ^ b) =< a
3 leo 158 . . . . . . 7 a =< (a v c)
42, 3letr 137 . . . . . 6 (a ^ b) =< (a v c)
54lecon 154 . . . . 5 (a v c)' =< (a ^ b)'
65leror 152 . . . 4 ((a v c)' v ((a ^ b) ^ c)) =< ((a ^ b)' v ((a ^ b) ^ c))
71, 6letr 137 . . 3 (((a v c)' v ((a ^ b) ^ c)) ^ (b ->1 c)) =< ((a ^ b)' v ((a ^ b) ^ c))
8 comanblem1 870 . . 3 ((a == c) ^ (b == c)) = (((a v c)' v ((a ^ b) ^ c)) ^ (b ->1 c))
9 df-i1 44 . . . 4 ((a ^ b) ->1 ((a == c) ^ (b == c))) = ((a ^ b)' v ((a ^ b) ^ ((a == c) ^ (b == c))))
10 comanblem2 871 . . . . 5 ((a ^ b) ^ ((a == c) ^ (b == c))) = ((a ^ b) ^ c)
1110lor 70 . . . 4 ((a ^ b)' v ((a ^ b) ^ ((a == c) ^ (b == c)))) = ((a ^ b)' v ((a ^ b) ^ c))
129, 11ax-r2 36 . . 3 ((a ^ b) ->1 ((a == c) ^ (b == c))) = ((a ^ b)' v ((a ^ b) ^ c))
137, 8, 12le3tr1 140 . 2 ((a == c) ^ (b == c)) =< ((a ^ b) ->1 ((a == c) ^ (b == c)))
1413i1com 708 1 (a ^ b) C ((a == c) ^ (b == c))
Colors of variables: term
Syntax hints:   C wc 3  'wn 4   == tb 5   v wo 6   ^ wa 7   ->1 wi1 12
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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  comanbn  873
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