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Theorem u5lem3 753
Description: Lemma for unified implication study.
Assertion
Ref Expression
u5lem3 (a ->5 (b ->5 a)) = (a' v ((a ^ b) v (a ^ b')))
Proof of Theorem u5lem3
StepHypRef Expression
1 u5lemc1b 685 . . 3 a C (b ->5 a)
21u5lemc4 705 . 2 (a ->5 (b ->5 a)) = (a' v (b ->5 a))
3 ax-a2 31 . . 3 (a' v (b ->5 a)) = ((b ->5 a) v a')
4 u5lemonb 639 . . . 4 ((b ->5 a) v a') = (((b ^ a) v (b' ^ a)) v a')
5 ancom 74 . . . . . . 7 (b ^ a) = (a ^ b)
6 ancom 74 . . . . . . 7 (b' ^ a) = (a ^ b')
75, 62or 72 . . . . . 6 ((b ^ a) v (b' ^ a)) = ((a ^ b) v (a ^ b'))
87ax-r5 38 . . . . 5 (((b ^ a) v (b' ^ a)) v a') = (((a ^ b) v (a ^ b')) v a')
9 ax-a2 31 . . . . 5 (((a ^ b) v (a ^ b')) v a') = (a' v ((a ^ b) v (a ^ b')))
108, 9ax-r2 36 . . . 4 (((b ^ a) v (b' ^ a)) v a') = (a' v ((a ^ b) v (a ^ b')))
114, 10ax-r2 36 . . 3 ((b ->5 a) v a') = (a' v ((a ^ b) v (a ^ b')))
123, 11ax-r2 36 . 2 (a' v (b ->5 a)) = (a' v ((a ^ b) v (a ^ b')))
132, 12ax-r2 36 1 (a ->5 (b ->5 a)) = (a' v ((a ^ b) v (a ^ b')))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->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:  u5lem3n  756  u5lem4  760
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