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Theorem u2lemc4 702
 Description: Lemma for Dishkant implication study.
Hypothesis
Ref Expression
ulemc3.1 a C b
Assertion
Ref Expression
u2lemc4 (a2 b) = (ab)

Proof of Theorem u2lemc4
StepHypRef Expression
1 df-i2 45 . 2 (a2 b) = (b ∪ (ab ))
2 ulemc3.1 . . . . 5 a C b
32comcom3 454 . . . 4 a C b
42comcom4 455 . . . 4 a C b
53, 4fh4 472 . . 3 (b ∪ (ab )) = ((ba ) ∩ (bb ))
6 ax-a2 31 . . . . 5 (ba ) = (ab)
7 df-t 41 . . . . . 6 1 = (bb )
87ax-r1 35 . . . . 5 (bb ) = 1
96, 82an 79 . . . 4 ((ba ) ∩ (bb )) = ((ab) ∩ 1)
10 an1 106 . . . 4 ((ab) ∩ 1) = (ab)
119, 10ax-r2 36 . . 3 ((ba ) ∩ (bb )) = (ab)
125, 11ax-r2 36 . 2 (b ∪ (ab )) = (ab)
131, 12ax-r2 36 1 (a2 b) = (ab)
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ 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  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:  u2lemle1  711
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