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Theorem negant2 858
 Description: Negated antecedent identity.
Hypothesis
Ref Expression
negant.1 (a1 c) = (b1 c)
Assertion
Ref Expression
negant2 (a2 c) = (b2 c)

Proof of Theorem negant2
StepHypRef Expression
1 negant.1 . . . . 5 (a1 c) = (b1 c)
21negantlem6 854 . . . 4 (ac ) = (bc )
3 ax-a1 30 . . . . 5 a = a
43ran 78 . . . 4 (ac ) = (a c )
5 ax-a1 30 . . . . 5 b = b
65ran 78 . . . 4 (bc ) = (b c )
72, 4, 63tr2 64 . . 3 (a c ) = (b c )
87lor 70 . 2 (c ∪ (a c )) = (c ∪ (b c ))
9 df-i2 45 . 2 (a2 c) = (c ∪ (a c ))
10 df-i2 45 . 2 (b2 c) = (c ∪ (b c ))
118, 9, 103tr1 63 1 (a2 c) = (b2 c)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ 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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