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Theorem u4lem3 752
 Description: Lemma for unified implication study. (Contributed by NM, 17-Dec-1997.)
Assertion
Ref Expression
u4lem3 (a4 (b4 a)) = (a ∪ ((ab) ∪ (ab )))

Proof of Theorem u4lem3
StepHypRef Expression
1 u4lemc1 683 . . 3 a C (b4 a)
21u4lemc4 704 . 2 (a4 (b4 a)) = (a ∪ (b4 a))
3 ax-a2 31 . . 3 (a ∪ (b4 a)) = ((b4 a) ∪ a )
4 u4lemonb 638 . . . 4 ((b4 a) ∪ a ) = (((ba) ∪ (ba)) ∪ a )
5 ancom 74 . . . . . . 7 (ba) = (ab)
6 ancom 74 . . . . . . 7 (ba) = (ab )
75, 62or 72 . . . . . 6 ((ba) ∪ (ba)) = ((ab) ∪ (ab ))
87ax-r5 38 . . . . 5 (((ba) ∪ (ba)) ∪ a ) = (((ab) ∪ (ab )) ∪ a )
9 ax-a2 31 . . . . 5 (((ab) ∪ (ab )) ∪ a ) = (a ∪ ((ab) ∪ (ab )))
108, 9ax-r2 36 . . . 4 (((ba) ∪ (ba)) ∪ a ) = (a ∪ ((ab) ∪ (ab )))
114, 10ax-r2 36 . . 3 ((b4 a) ∪ a ) = (a ∪ ((ab) ∪ (ab )))
123, 11ax-r2 36 . 2 (a ∪ (b4 a)) = (a ∪ ((ab) ∪ (ab )))
132, 12ax-r2 36 1 (a4 (b4 a)) = (a ∪ ((ab) ∪ (ab )))
 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-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:  u4lem3n  755  u4lem4  759
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