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Theorem neg3antlem1 864
 Description: Lemma for negated antecedent identity. (Contributed by NM, 7-Aug-2001.)
Hypothesis
Ref Expression
neg3ant.1 (a3 c) = (b3 c)
Assertion
Ref Expression
neg3antlem1 (ac) ≤ (b1 c)

Proof of Theorem neg3antlem1
StepHypRef Expression
1 leo 158 . . 3 (ac) ≤ ((ac) ∪ (ac))
2 neg3ant.1 . . . . . 6 (a3 c) = (b3 c)
32ran 78 . . . . 5 ((a3 c) ∩ c) = ((b3 c) ∩ c)
4 u3lemab 612 . . . . 5 ((a3 c) ∩ c) = ((ac) ∪ (ac))
5 u3lemab 612 . . . . 5 ((b3 c) ∩ c) = ((bc) ∪ (bc))
63, 4, 53tr2 64 . . . 4 ((ac) ∪ (ac)) = ((bc) ∪ (bc))
7 u1lemab 610 . . . . 5 ((b1 c) ∩ c) = ((bc) ∪ (bc))
87ax-r1 35 . . . 4 ((bc) ∪ (bc)) = ((b1 c) ∩ c)
96, 8ax-r2 36 . . 3 ((ac) ∪ (ac)) = ((b1 c) ∩ c)
101, 9lbtr 139 . 2 (ac) ≤ ((b1 c) ∩ c)
11 lea 160 . 2 ((b1 c) ∩ c) ≤ (b1 c)
1210, 11letr 137 1 (ac) ≤ (b1 c)
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2  ⊥ wn 4   ∪ 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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