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Theorem wql2lem5 292
 Description: Lemma for →2 WQL axiom.
Hypothesis
Ref Expression
wql2lem5.1 (a2 b) = 1
Assertion
Ref Expression
wql2lem5 ((b ∩ (ab)) →2 a ) = 1

Proof of Theorem wql2lem5
StepHypRef Expression
1 anor3 90 . . . 4 ((b ∩ (ab))a ) = ((b ∩ (ab)) ∪ a )
2 oran3 93 . . . . . 6 ((a2 b)a ) = ((a2 b) ∩ a)
3 ud2lem0c 278 . . . . . . 7 (a2 b) = (b ∩ (ab))
43ax-r5 38 . . . . . 6 ((a2 b)a ) = ((b ∩ (ab)) ∪ a )
5 wql2lem5.1 . . . . . . . . 9 (a2 b) = 1
65ran 78 . . . . . . . 8 ((a2 b) ∩ a) = (1 ∩ a)
7 ancom 74 . . . . . . . 8 (1 ∩ a) = (a ∩ 1)
8 an1 106 . . . . . . . 8 (a ∩ 1) = a
96, 7, 83tr 65 . . . . . . 7 ((a2 b) ∩ a) = a
109ax-r4 37 . . . . . 6 ((a2 b) ∩ a) = a
112, 4, 103tr2 64 . . . . 5 ((b ∩ (ab)) ∪ a ) = a
1211ax-r4 37 . . . 4 ((b ∩ (ab)) ∪ a ) = a
131, 12ax-r2 36 . . 3 ((b ∩ (ab))a ) = a
1413lor 70 . 2 (a ∪ ((b ∩ (ab))a )) = (aa )
15 df-i2 45 . 2 ((b ∩ (ab)) →2 a ) = (a ∪ ((b ∩ (ab))a ))
16 df-t 41 . 2 1 = (aa )
1714, 15, 163tr1 63 1 ((b ∩ (ab)) →2 a ) = 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-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 This theorem is referenced by: (None)
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