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Theorem imp3 841
Description: Implicational product with 3 variables. Theorem 3.20 of "Equations, states, and lattices..." paper.
Assertion
Ref Expression
imp3 ((a ->2 b) ^ (b ->1 c)) = ((a' ^ b') v (b ^ c))
Proof of Theorem imp3
StepHypRef Expression
1 df-i1 44 . . 3 (b ->1 c) = (b' v (b ^ c))
21lan 77 . 2 ((a ->2 b) ^ (b ->1 c)) = ((a ->2 b) ^ (b' v (b ^ c)))
3 u2lemc1 681 . . . 4 b C (a ->2 b)
43comcom3 454 . . 3 b' C (a ->2 b)
5 comanr1 464 . . . 4 b C (b ^ c)
65comcom3 454 . . 3 b' C (b ^ c)
74, 6fh2 470 . 2 ((a ->2 b) ^ (b' v (b ^ c))) = (((a ->2 b) ^ b') v ((a ->2 b) ^ (b ^ c)))
8 u2lemanb 616 . . 3 ((a ->2 b) ^ b') = (a' ^ b')
9 ancom 74 . . . 4 ((a ->2 b) ^ (b ^ c)) = ((b ^ c) ^ (a ->2 b))
10 lea 160 . . . . . 6 (b ^ c) =< b
11 u2lem3 750 . . . . . . 7 (b ->2 (a ->2 b)) = 1
1211u2lemle2 716 . . . . . 6 b =< (a ->2 b)
1310, 12letr 137 . . . . 5 (b ^ c) =< (a ->2 b)
1413df2le2 136 . . . 4 ((b ^ c) ^ (a ->2 b)) = (b ^ c)
159, 14ax-r2 36 . . 3 ((a ->2 b) ^ (b ^ c)) = (b ^ c)
168, 152or 72 . 2 (((a ->2 b) ^ b') v ((a ->2 b) ^ (b ^ c))) = ((a' ^ b') v (b ^ c))
172, 7, 163tr 65 1 ((a ->2 b) ^ (b ->1 c)) = ((a' ^ b') v (b ^ c))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->2 wi2 13
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-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  orbi  842  mlaconj4  844  mhcor1  888
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