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Theorem oaeqv 830
 Description: Weakened OA implies OA).
Hypothesis
Ref Expression
oaeqv.1 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ ((bc) →2 ((a2 b) ∩ (a2 c)))
Assertion
Ref Expression
oaeqv ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ (a2 c)

Proof of Theorem oaeqv
StepHypRef Expression
1 lea 160 . . . 4 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ (a2 b)
2 oaeqv.1 . . . 4 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ ((bc) →2 ((a2 b) ∩ (a2 c)))
31, 2ler2an 173 . . 3 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c))))
4 2oath1 826 . . 3 ((a2 b) ∩ ((bc) →2 ((a2 b) ∩ (a2 c)))) = ((a2 b) ∩ (a2 c))
53, 4lbtr 139 . 2 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ ((a2 b) ∩ (a2 c))
6 lear 161 . 2 ((a2 b) ∩ (a2 c)) ≤ (a2 c)
75, 6letr 137 1 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ (a2 c)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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