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Theorem u5lem5 765
Description: Lemma for unified implication study.
Assertion
Ref Expression
u5lem5 (a ->5 (a ->5 b)) = (a' v (a ^ b))
Proof of Theorem u5lem5
StepHypRef Expression
1 df-i5 48 . 2 (a ->5 (a ->5 b)) = (((a ^ (a ->5 b)) v (a' ^ (a ->5 b))) v (a' ^ (a ->5 b)'))
2 u5lemc1 684 . . . . . . . 8 a C (a ->5 b)
32comcom 453 . . . . . . 7 (a ->5 b) C a
43comcom2 183 . . . . . . 7 (a ->5 b) C a'
53, 4fh1r 473 . . . . . 6 ((a v a') ^ (a ->5 b)) = ((a ^ (a ->5 b)) v (a' ^ (a ->5 b)))
65ax-r1 35 . . . . 5 ((a ^ (a ->5 b)) v (a' ^ (a ->5 b))) = ((a v a') ^ (a ->5 b))
7 ancom 74 . . . . . 6 ((a v a') ^ (a ->5 b)) = ((a ->5 b) ^ (a v a'))
8 df-t 41 . . . . . . . . 9 1 = (a v a')
98ax-r1 35 . . . . . . . 8 (a v a') = 1
109lan 77 . . . . . . 7 ((a ->5 b) ^ (a v a')) = ((a ->5 b) ^ 1)
11 an1 106 . . . . . . 7 ((a ->5 b) ^ 1) = (a ->5 b)
1210, 11ax-r2 36 . . . . . 6 ((a ->5 b) ^ (a v a')) = (a ->5 b)
137, 12ax-r2 36 . . . . 5 ((a v a') ^ (a ->5 b)) = (a ->5 b)
146, 13ax-r2 36 . . . 4 ((a ^ (a ->5 b)) v (a' ^ (a ->5 b))) = (a ->5 b)
1514ax-r5 38 . . 3 (((a ^ (a ->5 b)) v (a' ^ (a ->5 b))) v (a' ^ (a ->5 b)')) = ((a ->5 b) v (a' ^ (a ->5 b)'))
162comcom3 454 . . . . 5 a' C (a ->5 b)
172comcom4 455 . . . . 5 a' C (a ->5 b)'
1816, 17fh4 472 . . . 4 ((a ->5 b) v (a' ^ (a ->5 b)')) = (((a ->5 b) v a') ^ ((a ->5 b) v (a ->5 b)'))
19 df-t 41 . . . . . . 7 1 = ((a ->5 b) v (a ->5 b)')
2019ax-r1 35 . . . . . 6 ((a ->5 b) v (a ->5 b)') = 1
2120lan 77 . . . . 5 (((a ->5 b) v a') ^ ((a ->5 b) v (a ->5 b)')) = (((a ->5 b) v a') ^ 1)
22 an1 106 . . . . . 6 (((a ->5 b) v a') ^ 1) = ((a ->5 b) v a')
23 u5lemona 629 . . . . . 6 ((a ->5 b) v a') = (a' v (a ^ b))
2422, 23ax-r2 36 . . . . 5 (((a ->5 b) v a') ^ 1) = (a' v (a ^ b))
2521, 24ax-r2 36 . . . 4 (((a ->5 b) v a') ^ ((a ->5 b) v (a ->5 b)')) = (a' v (a ^ b))
2618, 25ax-r2 36 . . 3 ((a ->5 b) v (a' ^ (a ->5 b)')) = (a' v (a ^ b))
2715, 26ax-r2 36 . 2 (((a ^ (a ->5 b)) v (a' ^ (a ->5 b))) v (a' ^ (a ->5 b)')) = (a' v (a ^ b))
281, 27ax-r2 36 1 (a ->5 (a ->5 b)) = (a' v (a ^ b))
Colors of variables: term
Syntax hints:   = wb 1  '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:  u5lem6  769
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