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Theorem negant2 858
Description: Negated antecedent identity.
Hypothesis
Ref Expression
negant.1 (a ->1 c) = (b ->1 c)
Assertion
Ref Expression
negant2 (a' ->2 c) = (b' ->2 c)
Proof of Theorem negant2
StepHypRef Expression
1 negant.1 . . . . 5 (a ->1 c) = (b ->1 c)
21negantlem6 854 . . . 4 (a ^ c') = (b ^ c')
3 ax-a1 30 . . . . 5 a = a''
43ran 78 . . . 4 (a ^ c') = (a'' ^ c')
5 ax-a1 30 . . . . 5 b = b''
65ran 78 . . . 4 (b ^ c') = (b'' ^ c')
72, 4, 63tr2 64 . . 3 (a'' ^ c') = (b'' ^ c')
87lor 70 . 2 (c v (a'' ^ c')) = (c v (b'' ^ c'))
9 df-i2 45 . 2 (a' ->2 c) = (c v (a'' ^ c'))
10 df-i2 45 . 2 (b' ->2 c) = (c v (b'' ^ c'))
118, 9, 103tr1 63 1 (a' ->2 c) = (b' ->2 c)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->2 wi2 13
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-i1 44  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  negant5  863
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