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Theorem ud4lem1d 580
 Description: Lemma for unified disjunction.
Assertion
Ref Expression
ud4lem1d (((a4 b) ∪ (b4 a)) ∩ (b4 a) ) = (((ab ) ∩ (ab)) ∩ a)

Proof of Theorem ud4lem1d
StepHypRef Expression
1 ud4lem1c 579 . . 3 ((a4 b) ∪ (b4 a)) = (ab )
2 ud4lem0c 280 . . 3 (b4 a) = (((ba ) ∩ (ba )) ∩ ((ba ) ∪ a))
31, 22an 79 . 2 (((a4 b) ∪ (b4 a)) ∩ (b4 a) ) = ((ab ) ∩ (((ba ) ∩ (ba )) ∩ ((ba ) ∪ a)))
4 an12 81 . . 3 ((ab ) ∩ (((ba ) ∩ (ba )) ∩ ((ba ) ∪ a))) = (((ba ) ∩ (ba )) ∩ ((ab ) ∩ ((ba ) ∪ a)))
5 ax-a2 31 . . . . 5 (ba ) = (ab )
6 ax-a2 31 . . . . 5 (ba ) = (ab)
75, 62an 79 . . . 4 ((ba ) ∩ (ba )) = ((ab ) ∩ (ab))
8 comor2 462 . . . . . . . . 9 (ab ) C b
98comcom3 454 . . . . . . . 8 (ab ) C b
109comcom5 458 . . . . . . 7 (ab ) C b
11 comor1 461 . . . . . . . 8 (ab ) C a
1211comcom2 183 . . . . . . 7 (ab ) C a
1310, 12com2an 484 . . . . . 6 (ab ) C (ba )
1413, 11fh1 469 . . . . 5 ((ab ) ∩ ((ba ) ∪ a)) = (((ab ) ∩ (ba )) ∪ ((ab ) ∩ a))
15 ax-a2 31 . . . . . . . . 9 (ab ) = (ba)
16 anor1 88 . . . . . . . . 9 (ba ) = (ba)
1715, 162an 79 . . . . . . . 8 ((ab ) ∩ (ba )) = ((ba) ∩ (ba) )
18 dff 101 . . . . . . . . 9 0 = ((ba) ∩ (ba) )
1918ax-r1 35 . . . . . . . 8 ((ba) ∩ (ba) ) = 0
2017, 19ax-r2 36 . . . . . . 7 ((ab ) ∩ (ba )) = 0
21 ancom 74 . . . . . . . 8 ((ab ) ∩ a) = (a ∩ (ab ))
22 anabs 121 . . . . . . . 8 (a ∩ (ab )) = a
2321, 22ax-r2 36 . . . . . . 7 ((ab ) ∩ a) = a
2420, 232or 72 . . . . . 6 (((ab ) ∩ (ba )) ∪ ((ab ) ∩ a)) = (0 ∪ a)
25 ax-a2 31 . . . . . . 7 (0 ∪ a) = (a ∪ 0)
26 or0 102 . . . . . . 7 (a ∪ 0) = a
2725, 26ax-r2 36 . . . . . 6 (0 ∪ a) = a
2824, 27ax-r2 36 . . . . 5 (((ab ) ∩ (ba )) ∪ ((ab ) ∩ a)) = a
2914, 28ax-r2 36 . . . 4 ((ab ) ∩ ((ba ) ∪ a)) = a
307, 292an 79 . . 3 (((ba ) ∩ (ba )) ∩ ((ab ) ∩ ((ba ) ∪ a))) = (((ab ) ∩ (ab)) ∩ a)
314, 30ax-r2 36 . 2 ((ab ) ∩ (((ba ) ∩ (ba )) ∩ ((ba ) ∪ a))) = (((ab ) ∩ (ab)) ∩ a)
323, 31ax-r2 36 1 (((a4 b) ∪ (b4 a)) ∩ (b4 a) ) = (((ab ) ∩ (ab)) ∩ a)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →4 wi4 15 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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  ud4lem1  581
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