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Theorem u3lem14a 791
 Description: Lemma for unified implication study. (Contributed by NM, 19-Jan-1998.)
Assertion
Ref Expression
u3lem14a (a3 ((b3 a ) →3 b )) = (a3 (b3 a))

Proof of Theorem u3lem14a
StepHypRef Expression
1 u3lem13b 790 . . 3 ((b3 a ) →3 b ) = (b1 a)
21ud3lem0a 260 . 2 (a3 ((b3 a ) →3 b )) = (a3 (b1 a))
3 df-i3 46 . . 3 (a3 (b1 a)) = (((a ∩ (b1 a)) ∪ (a ∩ (b1 a) )) ∪ (a ∩ (a ∪ (b1 a))))
4 ancom 74 . . . . . . . 8 (a ∩ (b1 a)) = ((b1 a) ∩ a )
5 u1lemanb 615 . . . . . . . 8 ((b1 a) ∩ a ) = (ba )
64, 5ax-r2 36 . . . . . . 7 (a ∩ (b1 a)) = (ba )
7 ancom 74 . . . . . . . 8 (a ∩ (b1 a) ) = ((b1 a)a )
8 u1lemnanb 655 . . . . . . . 8 ((b1 a)a ) = (ba )
97, 8ax-r2 36 . . . . . . 7 (a ∩ (b1 a) ) = (ba )
106, 92or 72 . . . . . 6 ((a ∩ (b1 a)) ∪ (a ∩ (b1 a) )) = ((ba ) ∪ (ba ))
11 ax-a2 31 . . . . . . 7 ((ba ) ∪ (ba )) = ((ba ) ∪ (ba ))
12 ancom 74 . . . . . . . 8 (ba ) = (ab)
13 ancom 74 . . . . . . . 8 (ba ) = (ab )
1412, 132or 72 . . . . . . 7 ((ba ) ∪ (ba )) = ((ab) ∪ (ab ))
1511, 14ax-r2 36 . . . . . 6 ((ba ) ∪ (ba )) = ((ab) ∪ (ab ))
1610, 15ax-r2 36 . . . . 5 ((a ∩ (b1 a)) ∪ (a ∩ (b1 a) )) = ((ab) ∪ (ab ))
17 ax-a2 31 . . . . . . . 8 (a ∪ (b1 a)) = ((b1 a) ∪ a )
18 u1lemonb 635 . . . . . . . 8 ((b1 a) ∪ a ) = 1
1917, 18ax-r2 36 . . . . . . 7 (a ∪ (b1 a)) = 1
2019lan 77 . . . . . 6 (a ∩ (a ∪ (b1 a))) = (a ∩ 1)
21 an1 106 . . . . . 6 (a ∩ 1) = a
2220, 21ax-r2 36 . . . . 5 (a ∩ (a ∪ (b1 a))) = a
2316, 222or 72 . . . 4 (((a ∩ (b1 a)) ∪ (a ∩ (b1 a) )) ∪ (a ∩ (a ∪ (b1 a)))) = (((ab) ∪ (ab )) ∪ a)
24 ax-a2 31 . . . . 5 (((ab) ∪ (ab )) ∪ a) = (a ∪ ((ab) ∪ (ab )))
25 u3lem3 751 . . . . . . 7 (a3 (b3 a)) = (a ∪ ((ab) ∪ (ab )))
2625ax-r1 35 . . . . . 6 (a ∪ ((ab) ∪ (ab ))) = (a3 (b3 a))
27 id 59 . . . . . 6 (a3 (b3 a)) = (a3 (b3 a))
2826, 27ax-r2 36 . . . . 5 (a ∪ ((ab) ∪ (ab ))) = (a3 (b3 a))
2924, 28ax-r2 36 . . . 4 (((ab) ∪ (ab )) ∪ a) = (a3 (b3 a))
3023, 29ax-r2 36 . . 3 (((a ∩ (b1 a)) ∪ (a ∩ (b1 a) )) ∪ (a ∩ (a ∪ (b1 a)))) = (a3 (b3 a))
313, 30ax-r2 36 . 2 (a3 (b1 a)) = (a3 (b3 a))
322, 31ax-r2 36 1 (a3 ((b3 a ) →3 b )) = (a3 (b3 a))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12   →3 wi3 14 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-i1 44  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u3lem14aa  792
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