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Theorem i3abs1 522
Description: Antecedent absorption.
Assertion
Ref Expression
i3abs1 (a ->3 (a ->3 (a ->3 b))) = (a ->3 (a ->3 b))
Proof of Theorem i3abs1
StepHypRef Expression
1 orordi 112 . . . . 5 (a' v ((a' ^ b) v (a' ^ b'))) = ((a' v (a' ^ b)) v (a' v (a' ^ b')))
2 orabs 120 . . . . . . 7 (a' v (a' ^ b)) = a'
3 orabs 120 . . . . . . 7 (a' v (a' ^ b')) = a'
42, 32or 72 . . . . . 6 ((a' v (a' ^ b)) v (a' v (a' ^ b'))) = (a' v a')
5 oridm 110 . . . . . 6 (a' v a') = a'
64, 5ax-r2 36 . . . . 5 ((a' v (a' ^ b)) v (a' v (a' ^ b'))) = a'
71, 6ax-r2 36 . . . 4 (a' v ((a' ^ b) v (a' ^ b'))) = a'
87ax-r5 38 . . 3 ((a' v ((a' ^ b) v (a' ^ b'))) v (a ^ (a' v b))) = (a' v (a ^ (a' v b)))
9 ax-a3 32 . . 3 ((a' v ((a' ^ b) v (a' ^ b'))) v (a ^ (a' v b))) = (a' v (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))))
10 omln 446 . . 3 (a' v (a ^ (a' v b))) = (a' v b)
118, 9, 103tr2 64 . 2 (a' v (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b)))) = (a' v b)
12 lem4 511 . . 3 (a ->3 (a ->3 (a ->3 b))) = (a' v (a ->3 b))
13 df-i3 46 . . . 4 (a ->3 b) = (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b)))
1413lor 70 . . 3 (a' v (a ->3 b)) = (a' v (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))))
1512, 14ax-r2 36 . 2 (a ->3 (a ->3 (a ->3 b))) = (a' v (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))))
16 lem4 511 . 2 (a ->3 (a ->3 b)) = (a' v b)
1711, 15, 163tr1 63 1 (a ->3 (a ->3 (a ->3 b))) = (a ->3 (a ->3 b))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->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:  i3abs2  523  i3th6  548
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