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Theorem di 126
Description: Dishkant implication.
Assertion
Ref Expression
di ((a ^ b) == a) = (a' v (a ^ b))
Proof of Theorem di
StepHypRef Expression
1 conb 122 . . 3 ((b' v a')' == a) = ((b' v a')'' == a')
2 ax-a1 30 . . . . . 6 (b' v a') = (b' v a')''
32ax-r1 35 . . . . 5 (b' v a')'' = (b' v a')
43rbi 98 . . . 4 ((b' v a')'' == a') = ((b' v a') == a')
5 mi 125 . . . 4 ((b' v a') == a') = (a' v (b'' ^ a''))
64, 5ax-r2 36 . . 3 ((b' v a')'' == a') = (a' v (b'' ^ a''))
71, 6ax-r2 36 . 2 ((b' v a')' == a) = (a' v (b'' ^ a''))
8 ancom 74 . . . 4 (a ^ b) = (b ^ a)
9 df-a 40 . . . 4 (b ^ a) = (b' v a')'
108, 9ax-r2 36 . . 3 (a ^ b) = (b' v a')'
1110rbi 98 . 2 ((a ^ b) == a) = ((b' v a')' == a)
12 ax-a1 30 . . . . 5 b = b''
13 ax-a1 30 . . . . 5 a = a''
1412, 132an 79 . . . 4 (b ^ a) = (b'' ^ a'')
158, 14ax-r2 36 . . 3 (a ^ b) = (b'' ^ a'')
1615lor 70 . 2 (a' v (a ^ b)) = (a' v (b'' ^ a''))
177, 11, 163tr1 63 1 ((a ^ b) == a) = (a' v (a ^ 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
This theorem is referenced by: (None)
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