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Theorem u1lemc4 701
 Description: Lemma for Sasaki implication study. (Contributed by NM, 24-Dec-1997.)
Hypothesis
Ref Expression
ulemc3.1 a C b
Assertion
Ref Expression
u1lemc4 (a1 b) = (ab)

Proof of Theorem u1lemc4
StepHypRef Expression
1 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
2 comid 187 . . . . 5 a C a
32comcom2 183 . . . 4 a C a
4 ulemc3.1 . . . 4 a C b
53, 4fh4 472 . . 3 (a ∪ (ab)) = ((aa) ∩ (ab))
6 ancom 74 . . . 4 ((aa) ∩ (ab)) = ((ab) ∩ (aa))
7 ax-a2 31 . . . . . . 7 (aa) = (aa )
8 df-t 41 . . . . . . . 8 1 = (aa )
98ax-r1 35 . . . . . . 7 (aa ) = 1
107, 9ax-r2 36 . . . . . 6 (aa) = 1
1110lan 77 . . . . 5 ((ab) ∩ (aa)) = ((ab) ∩ 1)
12 an1 106 . . . . 5 ((ab) ∩ 1) = (ab)
1311, 12ax-r2 36 . . . 4 ((ab) ∩ (aa)) = (ab)
146, 13ax-r2 36 . . 3 ((aa) ∩ (ab)) = (ab)
155, 14ax-r2 36 . 2 (a ∪ (ab)) = (ab)
161, 15ax-r2 36 1 (a1 b) = (ab)
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12 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-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by:  u1lemle1  710  u1lem1  734
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