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Theorem wql2lem 288
 Description: Classical implication inferred from Dishkant implication.
Hypothesis
Ref Expression
wql2lem.1 (a2 b) = 1
Assertion
Ref Expression
wql2lem (ab) = 1

Proof of Theorem wql2lem
StepHypRef Expression
1 le1 146 . 2 (ab) ≤ 1
2 df-i2 45 . . . 4 (a2 b) = (b ∪ (ab ))
3 wql2lem.1 . . . 4 (a2 b) = 1
4 ax-a2 31 . . . 4 (b ∪ (ab )) = ((ab ) ∪ b)
52, 3, 43tr2 64 . . 3 1 = ((ab ) ∪ b)
6 lea 160 . . . 4 (ab ) ≤ a
76leror 152 . . 3 ((ab ) ∪ b) ≤ (ab)
85, 7bltr 138 . 2 1 ≤ (ab)
91, 8lebi 145 1 (ab) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by:  wql2lem3  290
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