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Theorem biao 799
Description: Equivalence to biconditional.
Assertion
Ref Expression
biao (a == b) = ((a ^ b) == (a v b))
Proof of Theorem biao
StepHypRef Expression
1 leao1 162 . . . . 5 (a ^ b) =< (a v b)
21df2le2 136 . . . 4 ((a ^ b) ^ (a v b)) = (a ^ b)
32ax-r1 35 . . 3 (a ^ b) = ((a ^ b) ^ (a v b))
4 anor3 90 . . . 4 (a' ^ b') = (a v b)'
51lecon 154 . . . . . 6 (a v b)' =< (a ^ b)'
6 oridm 110 . . . . . . 7 ((a v b)' v (a v b)') = (a v b)'
76df-le1 130 . . . . . 6 (a v b)' =< (a v b)'
85, 7ler2an 173 . . . . 5 (a v b)' =< ((a ^ b)' ^ (a v b)')
9 lear 161 . . . . . . 7 ((a ^ b)' ^ (a v b)') =< (a v b)'
109df-le2 131 . . . . . 6 (((a ^ b)' ^ (a v b)') v (a v b)') = (a v b)'
1110df-le1 130 . . . . 5 ((a ^ b)' ^ (a v b)') =< (a v b)'
128, 11lebi 145 . . . 4 (a v b)' = ((a ^ b)' ^ (a v b)')
134, 12ax-r2 36 . . 3 (a' ^ b') = ((a ^ b)' ^ (a v b)')
143, 132or 72 . 2 ((a ^ b) v (a' ^ b')) = (((a ^ b) ^ (a v b)) v ((a ^ b)' ^ (a v b)'))
15 dfb 94 . 2 (a == b) = ((a ^ b) v (a' ^ b'))
16 dfb 94 . 2 ((a ^ b) == (a v b)) = (((a ^ b) ^ (a v b)) v ((a ^ b)' ^ (a v b)'))
1714, 15, 163tr1 63 1 (a == b) = ((a ^ b) == (a v b))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7
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
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131
This theorem is referenced by:  mlaconj4  844
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