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Theorem i3lem3 506
Description: Lemma for Kalmbach implication.
Hypothesis
Ref Expression
i3lem.1 (a ->3 b) = 1
Assertion
Ref Expression
i3lem3 ((a' v b) ^ b') = (a' ^ b')
Proof of Theorem i3lem3
StepHypRef Expression
1 omlan 448 . 2 (b' ^ (b v (b' ^ a'))) = (b' ^ a')
2 ancom 74 . . 3 ((a' v b) ^ b') = (b' ^ (a' v b))
3 ax-a2 31 . . . . 5 (a' v b) = (b v a')
4 ax-a3 32 . . . . . . 7 ((b v (a' ^ b)) v (a' ^ b')) = (b v ((a' ^ b) v (a' ^ b')))
54ax-r1 35 . . . . . 6 (b v ((a' ^ b) v (a' ^ b'))) = ((b v (a' ^ b)) v (a' ^ b'))
6 i3lem.1 . . . . . . . 8 (a ->3 b) = 1
76i3lem1 504 . . . . . . 7 ((a' ^ b) v (a' ^ b')) = a'
87lor 70 . . . . . 6 (b v ((a' ^ b) v (a' ^ b'))) = (b v a')
9 ancom 74 . . . . . . . . 9 (a' ^ b) = (b ^ a')
109lor 70 . . . . . . . 8 (b v (a' ^ b)) = (b v (b ^ a'))
11 orabs 120 . . . . . . . 8 (b v (b ^ a')) = b
1210, 11ax-r2 36 . . . . . . 7 (b v (a' ^ b)) = b
13 ancom 74 . . . . . . 7 (a' ^ b') = (b' ^ a')
1412, 132or 72 . . . . . 6 ((b v (a' ^ b)) v (a' ^ b')) = (b v (b' ^ a'))
155, 8, 143tr2 64 . . . . 5 (b v a') = (b v (b' ^ a'))
163, 15ax-r2 36 . . . 4 (a' v b) = (b v (b' ^ a'))
1716lan 77 . . 3 (b' ^ (a' v b)) = (b' ^ (b v (b' ^ a')))
182, 17ax-r2 36 . 2 ((a' v b) ^ b') = (b' ^ (b v (b' ^ a')))
191, 18, 133tr1 63 1 ((a' v b) ^ b') = (a' ^ b')
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->3 wi3 14
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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  i3le  515
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