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Theorem u3lem7 774
Description: Lemma for unified implication study.
Assertion
Ref Expression
u3lem7 (a ->3 (a' ->3 b)) = (a' v ((a ^ b) v (a ^ b')))
Proof of Theorem u3lem7
StepHypRef Expression
1 comi31 508 . . . 4 a' C (a' ->3 b)
21comcom6 459 . . 3 a C (a' ->3 b)
32u3lemc4 703 . 2 (a ->3 (a' ->3 b)) = (a' v (a' ->3 b))
4 df-i3 46 . . . 4 (a' ->3 b) = (((a'' ^ b) v (a'' ^ b')) v (a' ^ (a'' v b)))
54lor 70 . . 3 (a' v (a' ->3 b)) = (a' v (((a'' ^ b) v (a'' ^ b')) v (a' ^ (a'' v b))))
6 or12 80 . . . 4 (a' v (((a'' ^ b) v (a'' ^ b')) v (a' ^ (a'' v b)))) = (((a'' ^ b) v (a'' ^ b')) v (a' v (a' ^ (a'' v b))))
7 ax-a1 30 . . . . . . . . 9 a = a''
87ran 78 . . . . . . . 8 (a ^ b) = (a'' ^ b)
97ran 78 . . . . . . . 8 (a ^ b') = (a'' ^ b')
108, 92or 72 . . . . . . 7 ((a ^ b) v (a ^ b')) = ((a'' ^ b) v (a'' ^ b'))
1110ax-r1 35 . . . . . 6 ((a'' ^ b) v (a'' ^ b')) = ((a ^ b) v (a ^ b'))
12 orabs 120 . . . . . 6 (a' v (a' ^ (a'' v b))) = a'
1311, 122or 72 . . . . 5 (((a'' ^ b) v (a'' ^ b')) v (a' v (a' ^ (a'' v b)))) = (((a ^ b) v (a ^ b')) v a')
14 ax-a2 31 . . . . 5 (((a ^ b) v (a ^ b')) v a') = (a' v ((a ^ b) v (a ^ b')))
1513, 14ax-r2 36 . . . 4 (((a'' ^ b) v (a'' ^ b')) v (a' v (a' ^ (a'' v b)))) = (a' v ((a ^ b) v (a ^ b')))
166, 15ax-r2 36 . . 3 (a' v (((a'' ^ b) v (a'' ^ b')) v (a' ^ (a'' v b)))) = (a' v ((a ^ b) v (a ^ b')))
175, 16ax-r2 36 . 2 (a' v (a' ->3 b)) = (a' v ((a ^ b) v (a ^ b')))
183, 17ax-r2 36 1 (a ->3 (a' ->3 b)) = (a' v ((a ^ b) 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:  u3lem8  783  u3lem9  784
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