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Theorem oa3-3lem 981
Description: Lemma for 3-OA(3). Equivalence with substitution into 6-OA dual.
Assertion
Ref Expression
oa3-3lem (a' ^ (a v (b ^ (((a ^ b) v (a' ^ b')) v (((a ^ 1) v (a' ^ c)) ^ ((b ^ 1) v (b' ^ c))))))) = (a' ^ (a v (b ^ ((a == b) v ((a' ->1 c) ^ (b' ->1 c))))))
Proof of Theorem oa3-3lem
StepHypRef Expression
1 dfb 94 . . . . . 6 (a == b) = ((a ^ b) v (a' ^ b'))
21ax-r1 35 . . . . 5 ((a ^ b) v (a' ^ b')) = (a == b)
3 an1 106 . . . . . . . . 9 (a ^ 1) = a
4 ax-a1 30 . . . . . . . . 9 a = a''
53, 4ax-r2 36 . . . . . . . 8 (a ^ 1) = a''
65ax-r5 38 . . . . . . 7 ((a ^ 1) v (a' ^ c)) = (a'' v (a' ^ c))
7 df-i1 44 . . . . . . . 8 (a' ->1 c) = (a'' v (a' ^ c))
87ax-r1 35 . . . . . . 7 (a'' v (a' ^ c)) = (a' ->1 c)
96, 8ax-r2 36 . . . . . 6 ((a ^ 1) v (a' ^ c)) = (a' ->1 c)
10 an1 106 . . . . . . . . 9 (b ^ 1) = b
11 ax-a1 30 . . . . . . . . 9 b = b''
1210, 11ax-r2 36 . . . . . . . 8 (b ^ 1) = b''
1312ax-r5 38 . . . . . . 7 ((b ^ 1) v (b' ^ c)) = (b'' v (b' ^ c))
14 df-i1 44 . . . . . . . 8 (b' ->1 c) = (b'' v (b' ^ c))
1514ax-r1 35 . . . . . . 7 (b'' v (b' ^ c)) = (b' ->1 c)
1613, 15ax-r2 36 . . . . . 6 ((b ^ 1) v (b' ^ c)) = (b' ->1 c)
179, 162an 79 . . . . 5 (((a ^ 1) v (a' ^ c)) ^ ((b ^ 1) v (b' ^ c))) = ((a' ->1 c) ^ (b' ->1 c))
182, 172or 72 . . . 4 (((a ^ b) v (a' ^ b')) v (((a ^ 1) v (a' ^ c)) ^ ((b ^ 1) v (b' ^ c)))) = ((a == b) v ((a' ->1 c) ^ (b' ->1 c)))
1918lan 77 . . 3 (b ^ (((a ^ b) v (a' ^ b')) v (((a ^ 1) v (a' ^ c)) ^ ((b ^ 1) v (b' ^ c))))) = (b ^ ((a == b) v ((a' ->1 c) ^ (b' ->1 c))))
2019lor 70 . 2 (a v (b ^ (((a ^ b) v (a' ^ b')) v (((a ^ 1) v (a' ^ c)) ^ ((b ^ 1) v (b' ^ c)))))) = (a v (b ^ ((a == b) v ((a' ->1 c) ^ (b' ->1 c)))))
2120lan 77 1 (a' ^ (a v (b ^ (((a ^ b) v (a' ^ b')) v (((a ^ 1) v (a' ^ c)) ^ ((b ^ 1) v (b' ^ c))))))) = (a' ^ (a v (b ^ ((a == b) v ((a' ->1 c) ^ (b' ->1 c))))))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  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-i1 44
This theorem is referenced by:  oa3-6to3  987
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