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Theorem neg3antlem1 864
Description: Lemma for negated antecedent identity.
Hypothesis
Ref Expression
neg3ant.1 (a ->3 c) = (b ->3 c)
Assertion
Ref Expression
neg3antlem1 (a ^ c) =< (b ->1 c)
Proof of Theorem neg3antlem1
StepHypRef Expression
1 leo 158 . . 3 (a ^ c) =< ((a ^ c) v (a' ^ c))
2 neg3ant.1 . . . . . 6 (a ->3 c) = (b ->3 c)
32ran 78 . . . . 5 ((a ->3 c) ^ c) = ((b ->3 c) ^ c)
4 u3lemab 612 . . . . 5 ((a ->3 c) ^ c) = ((a ^ c) v (a' ^ c))
5 u3lemab 612 . . . . 5 ((b ->3 c) ^ c) = ((b ^ c) v (b' ^ c))
63, 4, 53tr2 64 . . . 4 ((a ^ c) v (a' ^ c)) = ((b ^ c) v (b' ^ c))
7 u1lemab 610 . . . . 5 ((b ->1 c) ^ c) = ((b ^ c) v (b' ^ c))
87ax-r1 35 . . . 4 ((b ^ c) v (b' ^ c)) = ((b ->1 c) ^ c)
96, 8ax-r2 36 . . 3 ((a ^ c) v (a' ^ c)) = ((b ->1 c) ^ c)
101, 9lbtr 139 . 2 (a ^ c) =< ((b ->1 c) ^ c)
11 lea 160 . 2 ((b ->1 c) ^ c) =< (b ->1 c)
1210, 11letr 137 1 (a ^ c) =< (b ->1 c)
Colors of variables: term
Syntax hints:   = wb 1   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->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-i1 44  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  neg3ant1  866
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