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Theorem u4lemc4 704
 Description: Lemma for non-tollens implication study.
Hypothesis
Ref Expression
ulemc3.1 a C b
Assertion
Ref Expression
u4lemc4 (a4 b) = (ab)

Proof of Theorem u4lemc4
StepHypRef Expression
1 df-i4 47 . 2 (a4 b) = (((ab) ∪ (ab)) ∪ ((ab) ∩ b ))
2 ulemc3.1 . . . . . . 7 a C b
3 comid 187 . . . . . . . 8 a C a
43comcom2 183 . . . . . . 7 a C a
52, 4fh2r 474 . . . . . 6 ((aa ) ∩ b) = ((ab) ∪ (ab))
65ax-r1 35 . . . . 5 ((ab) ∪ (ab)) = ((aa ) ∩ b)
7 ancom 74 . . . . . 6 ((aa ) ∩ b) = (b ∩ (aa ))
8 df-t 41 . . . . . . . . 9 1 = (aa )
98ax-r1 35 . . . . . . . 8 (aa ) = 1
109lan 77 . . . . . . 7 (b ∩ (aa )) = (b ∩ 1)
11 an1 106 . . . . . . 7 (b ∩ 1) = b
1210, 11ax-r2 36 . . . . . 6 (b ∩ (aa )) = b
137, 12ax-r2 36 . . . . 5 ((aa ) ∩ b) = b
146, 13ax-r2 36 . . . 4 ((ab) ∪ (ab)) = b
152comcom4 455 . . . . . 6 a C b
162comcom3 454 . . . . . 6 a C b
1715, 16fh2r 474 . . . . 5 ((ab) ∩ b ) = ((ab ) ∪ (bb ))
18 dff 101 . . . . . . . 8 0 = (bb )
1918ax-r1 35 . . . . . . 7 (bb ) = 0
2019lor 70 . . . . . 6 ((ab ) ∪ (bb )) = ((ab ) ∪ 0)
21 or0 102 . . . . . 6 ((ab ) ∪ 0) = (ab )
2220, 21ax-r2 36 . . . . 5 ((ab ) ∪ (bb )) = (ab )
2317, 22ax-r2 36 . . . 4 ((ab) ∩ b ) = (ab )
2414, 232or 72 . . 3 (((ab) ∪ (ab)) ∪ ((ab) ∩ b )) = (b ∪ (ab ))
2516, 15fh4 472 . . . 4 (b ∪ (ab )) = ((ba ) ∩ (bb ))
26 ax-a2 31 . . . . . 6 (ba ) = (ab)
27 df-t 41 . . . . . . 7 1 = (bb )
2827ax-r1 35 . . . . . 6 (bb ) = 1
2926, 282an 79 . . . . 5 ((ba ) ∩ (bb )) = ((ab) ∩ 1)
30 an1 106 . . . . 5 ((ab) ∩ 1) = (ab)
3129, 30ax-r2 36 . . . 4 ((ba ) ∩ (bb )) = (ab)
3225, 31ax-r2 36 . . 3 (b ∪ (ab )) = (ab)
3324, 32ax-r2 36 . 2 (((ab) ∪ (ab)) ∪ ((ab) ∩ b )) = (ab)
341, 33ax-r2 36 1 (a4 b) = (ab)
 Colors of variables: term Syntax hints:   = wb 1   C wc 3  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  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:  u4lemle1  713  u4lem2  747  u4lem3  752
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