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Theorem oa3-4lem 983
Description: Lemma for 3-OA(4). Equivalence with substitution into 6-OA dual.
Assertion
Ref Expression
oa3-4lem (a' ^ (a v (b ^ (((a ^ b) v (a' ^ b')) v (((a ^ c) v (a' ^ 1)) ^ ((b ^ c) v (b' ^ 1))))))) = (a' ^ (a v (b ^ ((a == b) v ((a ->1 c) ^ (b ->1 c))))))
Proof of Theorem oa3-4lem
StepHypRef Expression
1 dfb 94 . . . . . 6 (a == b) = ((a ^ b) v (a' ^ b'))
2 ax-a2 31 . . . . . . . 8 (a' v (a ^ c)) = ((a ^ c) v a')
3 df-i1 44 . . . . . . . 8 (a ->1 c) = (a' v (a ^ c))
4 an1 106 . . . . . . . . 9 (a' ^ 1) = a'
54lor 70 . . . . . . . 8 ((a ^ c) v (a' ^ 1)) = ((a ^ c) v a')
62, 3, 53tr1 63 . . . . . . 7 (a ->1 c) = ((a ^ c) v (a' ^ 1))
7 ax-a2 31 . . . . . . . 8 (b' v (b ^ c)) = ((b ^ c) v b')
8 df-i1 44 . . . . . . . 8 (b ->1 c) = (b' v (b ^ c))
9 an1 106 . . . . . . . . 9 (b' ^ 1) = b'
109lor 70 . . . . . . . 8 ((b ^ c) v (b' ^ 1)) = ((b ^ c) v b')
117, 8, 103tr1 63 . . . . . . 7 (b ->1 c) = ((b ^ c) v (b' ^ 1))
126, 112an 79 . . . . . 6 ((a ->1 c) ^ (b ->1 c)) = (((a ^ c) v (a' ^ 1)) ^ ((b ^ c) v (b' ^ 1)))
131, 122or 72 . . . . 5 ((a == b) v ((a ->1 c) ^ (b ->1 c))) = (((a ^ b) v (a' ^ b')) v (((a ^ c) v (a' ^ 1)) ^ ((b ^ c) v (b' ^ 1))))
1413ax-r1 35 . . . 4 (((a ^ b) v (a' ^ b')) v (((a ^ c) v (a' ^ 1)) ^ ((b ^ c) v (b' ^ 1)))) = ((a == b) v ((a ->1 c) ^ (b ->1 c)))
1514lan 77 . . 3 (b ^ (((a ^ b) v (a' ^ b')) v (((a ^ c) v (a' ^ 1)) ^ ((b ^ c) v (b' ^ 1))))) = (b ^ ((a == b) v ((a ->1 c) ^ (b ->1 c))))
1615lor 70 . 2 (a v (b ^ (((a ^ b) v (a' ^ b')) v (((a ^ c) v (a' ^ 1)) ^ ((b ^ c) v (b' ^ 1)))))) = (a v (b ^ ((a == b) v ((a ->1 c) ^ (b ->1 c)))))
1716lan 77 1 (a' ^ (a v (b ^ (((a ^ b) v (a' ^ b')) v (((a ^ c) v (a' ^ 1)) ^ ((b ^ c) v (b' ^ 1))))))) = (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:  oa3-2to4  988
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