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Theorem i3n2 501
Description: Equivalence for Kalmbach implication.
Assertion
Ref Expression
i3n2 (a' ->3 b') = ((a ^ b) v ((a v b') ^ (a' v (a ^ b'))))
Proof of Theorem i3n2
StepHypRef Expression
1 df2i3 498 . 2 (a' ->3 b') = ((a'' ^ b'') v ((a'' v b') ^ (a' v (a'' ^ b'))))
2 ax-a1 30 . . . . 5 a = a''
3 ax-a1 30 . . . . 5 b = b''
42, 32an 79 . . . 4 (a ^ b) = (a'' ^ b'')
52ax-r5 38 . . . . 5 (a v b') = (a'' v b')
62ran 78 . . . . . 6 (a ^ b') = (a'' ^ b')
76lor 70 . . . . 5 (a' v (a ^ b')) = (a' v (a'' ^ b'))
85, 72an 79 . . . 4 ((a v b') ^ (a' v (a ^ b'))) = ((a'' v b') ^ (a' v (a'' ^ b')))
94, 82or 72 . . 3 ((a ^ b) v ((a v b') ^ (a' v (a ^ b')))) = ((a'' ^ b'') v ((a'' v b') ^ (a' v (a'' ^ b'))))
109ax-r1 35 . 2 ((a'' ^ b'') v ((a'' v b') ^ (a' v (a'' ^ b')))) = ((a ^ b) v ((a v b') ^ (a' v (a ^ b'))))
111, 10ax-r2 36 1 (a' ->3 b') = ((a ^ b) v ((a v b') ^ (a' v (a ^ b'))))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->3 wi3 14
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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by: (None)
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