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Theorem ud4lem0c 280
 Description: Lemma for unified disjunction.
Assertion
Ref Expression
ud4lem0c (a4 b) = (((ab ) ∩ (ab )) ∩ ((ab ) ∪ b))

Proof of Theorem ud4lem0c
StepHypRef Expression
1 df-i4 47 . . 3 (a4 b) = (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))
2 oran 87 . . . 4 (((ab) ∪ (ab)) ∪ ((ab) ∩ b )) = (((ab) ∪ (ab)) ∩ ((ab) ∩ b ) )
3 oran 87 . . . . . . . 8 ((ab) ∪ (ab)) = ((ab) ∩ (ab) )
4 df-a 40 . . . . . . . . . . 11 (ab) = (ab )
54con2 67 . . . . . . . . . 10 (ab) = (ab )
6 anor2 89 . . . . . . . . . . 11 (ab) = (ab )
76con2 67 . . . . . . . . . 10 (ab) = (ab )
85, 72an 79 . . . . . . . . 9 ((ab) ∩ (ab) ) = ((ab ) ∩ (ab ))
98ax-r4 37 . . . . . . . 8 ((ab) ∩ (ab) ) = ((ab ) ∩ (ab ))
103, 9ax-r2 36 . . . . . . 7 ((ab) ∪ (ab)) = ((ab ) ∩ (ab ))
1110con2 67 . . . . . 6 ((ab) ∪ (ab)) = ((ab ) ∩ (ab ))
12 anor1 88 . . . . . . . 8 ((ab) ∩ b ) = ((ab)b)
13 anor1 88 . . . . . . . . . . 11 (ab ) = (ab)
1413ax-r1 35 . . . . . . . . . 10 (ab) = (ab )
1514ax-r5 38 . . . . . . . . 9 ((ab)b) = ((ab ) ∪ b)
1615ax-r4 37 . . . . . . . 8 ((ab)b) = ((ab ) ∪ b)
1712, 16ax-r2 36 . . . . . . 7 ((ab) ∩ b ) = ((ab ) ∪ b)
1817con2 67 . . . . . 6 ((ab) ∩ b ) = ((ab ) ∪ b)
1911, 182an 79 . . . . 5 (((ab) ∪ (ab)) ∩ ((ab) ∩ b ) ) = (((ab ) ∩ (ab )) ∩ ((ab ) ∪ b))
2019ax-r4 37 . . . 4 (((ab) ∪ (ab)) ∩ ((ab) ∩ b ) ) = (((ab ) ∩ (ab )) ∩ ((ab ) ∪ b))
212, 20ax-r2 36 . . 3 (((ab) ∪ (ab)) ∪ ((ab) ∩ b )) = (((ab ) ∩ (ab )) ∩ ((ab ) ∪ b))
221, 21ax-r2 36 . 2 (a4 b) = (((ab ) ∩ (ab )) ∩ ((ab ) ∪ b))
2322con2 67 1 (a4 b) = (((ab ) ∩ (ab )) ∩ ((ab ) ∪ b))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →4 wi4 15 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-a 40  df-i4 47 This theorem is referenced by:  ud4lem1b  578  ud4lem1c  579  ud4lem1d  580  ud4lem3a  583  ud4lem3b  584  u4lem5  764
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