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Theorem neg3ant1 866
 Description: Lemma for negated antecedent identity.
Hypothesis
Ref Expression
neg3ant.1 (a3 c) = (b3 c)
Assertion
Ref Expression
neg3ant1 (a1 c) = (b1 c)

Proof of Theorem neg3ant1
StepHypRef Expression
1 neg3ant.1 . . . . . 6 (a3 c) = (b3 c)
21neg3antlem2 865 . . . . 5 a ≤ (b1 c)
31neg3antlem1 864 . . . . 5 (ac) ≤ (b1 c)
42, 3lel2or 170 . . . 4 (a ∪ (ac)) ≤ (b1 c)
5 df-i1 44 . . . 4 (b1 c) = (b ∪ (bc))
64, 5lbtr 139 . . 3 (a ∪ (ac)) ≤ (b ∪ (bc))
71ax-r1 35 . . . . . 6 (b3 c) = (a3 c)
87neg3antlem2 865 . . . . 5 b ≤ (a1 c)
97neg3antlem1 864 . . . . 5 (bc) ≤ (a1 c)
108, 9lel2or 170 . . . 4 (b ∪ (bc)) ≤ (a1 c)
11 df-i1 44 . . . 4 (a1 c) = (a ∪ (ac))
1210, 11lbtr 139 . . 3 (b ∪ (bc)) ≤ (a ∪ (ac))
136, 12lebi 145 . 2 (a ∪ (ac)) = (b ∪ (bc))
1413, 11, 53tr1 63 1 (a1 c) = (b1 c)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ 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: (None)
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