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Theorem u2lem1 735
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u2lem1 ((a2 b) →2 a) = a

Proof of Theorem u2lem1
StepHypRef Expression
1 df-i2 45 . 2 ((a2 b) →2 a) = (a ∪ ((a2 b)a ))
2 ud2lem0c 278 . . . . . 6 (a2 b) = (b ∩ (ab))
32ran 78 . . . . 5 ((a2 b)a ) = ((b ∩ (ab)) ∩ a )
4 an32 83 . . . . . 6 ((b ∩ (ab)) ∩ a ) = ((ba ) ∩ (ab))
5 ax-a2 31 . . . . . . . . 9 (ab) = (ba)
6 oran 87 . . . . . . . . 9 (ba) = (ba )
75, 6ax-r2 36 . . . . . . . 8 (ab) = (ba )
87lan 77 . . . . . . 7 ((ba ) ∩ (ab)) = ((ba ) ∩ (ba ) )
9 dff 101 . . . . . . . 8 0 = ((ba ) ∩ (ba ) )
109ax-r1 35 . . . . . . 7 ((ba ) ∩ (ba ) ) = 0
118, 10ax-r2 36 . . . . . 6 ((ba ) ∩ (ab)) = 0
124, 11ax-r2 36 . . . . 5 ((b ∩ (ab)) ∩ a ) = 0
133, 12ax-r2 36 . . . 4 ((a2 b)a ) = 0
1413lor 70 . . 3 (a ∪ ((a2 b)a )) = (a ∪ 0)
15 or0 102 . . 3 (a ∪ 0) = a
1614, 15ax-r2 36 . 2 (a ∪ ((a2 b)a )) = a
171, 16ax-r2 36 1 ((a2 b) →2 a) = a
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →2 wi2 13 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  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:  u2lem1n  740
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