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Theorem u5lemc4 705
Description: Lemma for relevance implication study.
Hypothesis
Ref Expression
ulemc3.1 a C b
Assertion
Ref Expression
u5lemc4 (a ->5 b) = (a' v b)
Proof of Theorem u5lemc4
StepHypRef Expression
1 df-i5 48 . 2 (a ->5 b) = (((a ^ b) v (a' ^ b)) v (a' ^ b'))
2 ulemc3.1 . . . . . . 7 a C b
3 comid 187 . . . . . . . 8 a C a
43comcom2 183 . . . . . . 7 a C a'
52, 4fh2r 474 . . . . . 6 ((a v a') ^ b) = ((a ^ b) v (a' ^ b))
65ax-r1 35 . . . . 5 ((a ^ b) v (a' ^ b)) = ((a v a') ^ b)
7 ancom 74 . . . . . 6 ((a v a') ^ b) = (b ^ (a v a'))
8 df-t 41 . . . . . . . . 9 1 = (a v a')
98ax-r1 35 . . . . . . . 8 (a v a') = 1
109lan 77 . . . . . . 7 (b ^ (a v a')) = (b ^ 1)
11 an1 106 . . . . . . 7 (b ^ 1) = b
1210, 11ax-r2 36 . . . . . 6 (b ^ (a v a')) = b
137, 12ax-r2 36 . . . . 5 ((a v a') ^ b) = b
146, 13ax-r2 36 . . . 4 ((a ^ b) v (a' ^ b)) = b
1514ax-r5 38 . . 3 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = (b v (a' ^ b'))
162comcom3 454 . . . . 5 a' C b
172comcom4 455 . . . . 5 a' C b'
1816, 17fh4 472 . . . 4 (b v (a' ^ b')) = ((b v a') ^ (b v b'))
19 ax-a2 31 . . . . . 6 (b v a') = (a' v b)
20 df-t 41 . . . . . . 7 1 = (b v b')
2120ax-r1 35 . . . . . 6 (b v b') = 1
2219, 212an 79 . . . . 5 ((b v a') ^ (b v b')) = ((a' v b) ^ 1)
23 an1 106 . . . . 5 ((a' v b) ^ 1) = (a' v b)
2422, 23ax-r2 36 . . . 4 ((b v a') ^ (b v b')) = (a' v b)
2518, 24ax-r2 36 . . 3 (b v (a' ^ b')) = (a' v b)
2615, 25ax-r2 36 . 2 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = (a' v b)
271, 26ax-r2 36 1 (a ->5 b) = (a' v b)
Colors of variables: term
Syntax hints:   = wb 1   C wc 3  'wn 4   v wo 6   ^ wa 7  1wt 8   ->5 wi5 16
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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  u5lemle1  714  u5lem1  738  u5lem2  748  u5lem3  753  u5lem4  760
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