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Theorem u4lem4 759
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u4lem4 (a4 (a4 (b4 a))) = (a4 (b4 a))

Proof of Theorem u4lem4
StepHypRef Expression
1 df-i4 47 . 2 (a4 (a4 (b4 a))) = (((a ∩ (a4 (b4 a))) ∪ (a ∩ (a4 (b4 a)))) ∪ ((a ∪ (a4 (b4 a))) ∩ (a4 (b4 a)) ))
2 u4lem3 752 . . . . . . . . 9 (a4 (b4 a)) = (a ∪ ((ab) ∪ (ab )))
3 comid 187 . . . . . . . . . . . 12 a C a
43comcom2 183 . . . . . . . . . . 11 a C a
5 comanr1 464 . . . . . . . . . . . 12 a C (ab)
6 comanr1 464 . . . . . . . . . . . 12 a C (ab )
75, 6com2or 483 . . . . . . . . . . 11 a C ((ab) ∪ (ab ))
84, 7com2or 483 . . . . . . . . . 10 a C (a ∪ ((ab) ∪ (ab )))
98comcom 453 . . . . . . . . 9 (a ∪ ((ab) ∪ (ab ))) C a
102, 9bctr 181 . . . . . . . 8 (a4 (b4 a)) C a
1110comcom 453 . . . . . . 7 a C (a4 (b4 a))
1211, 4fh2r 474 . . . . . 6 ((aa ) ∩ (a4 (b4 a))) = ((a ∩ (a4 (b4 a))) ∪ (a ∩ (a4 (b4 a))))
1312ax-r1 35 . . . . 5 ((a ∩ (a4 (b4 a))) ∪ (a ∩ (a4 (b4 a)))) = ((aa ) ∩ (a4 (b4 a)))
14 ancom 74 . . . . . 6 ((aa ) ∩ (a4 (b4 a))) = ((a4 (b4 a)) ∩ (aa ))
15 df-t 41 . . . . . . . . 9 1 = (aa )
1615ax-r1 35 . . . . . . . 8 (aa ) = 1
1716lan 77 . . . . . . 7 ((a4 (b4 a)) ∩ (aa )) = ((a4 (b4 a)) ∩ 1)
18 an1 106 . . . . . . 7 ((a4 (b4 a)) ∩ 1) = (a4 (b4 a))
1917, 18ax-r2 36 . . . . . 6 ((a4 (b4 a)) ∩ (aa )) = (a4 (b4 a))
2014, 19ax-r2 36 . . . . 5 ((aa ) ∩ (a4 (b4 a))) = (a4 (b4 a))
2113, 20ax-r2 36 . . . 4 ((a ∩ (a4 (b4 a))) ∪ (a ∩ (a4 (b4 a)))) = (a4 (b4 a))
2210comcom4 455 . . . . . 6 (a4 (b4 a)) C a
23 comid 187 . . . . . . 7 (a4 (b4 a)) C (a4 (b4 a))
2423comcom3 454 . . . . . 6 (a4 (b4 a)) C (a4 (b4 a))
2522, 24fh1r 473 . . . . 5 ((a ∪ (a4 (b4 a))) ∩ (a4 (b4 a)) ) = ((a ∩ (a4 (b4 a)) ) ∪ ((a4 (b4 a)) ∩ (a4 (b4 a)) ))
26 dff 101 . . . . . . . 8 0 = ((a4 (b4 a)) ∩ (a4 (b4 a)) )
2726ax-r1 35 . . . . . . 7 ((a4 (b4 a)) ∩ (a4 (b4 a)) ) = 0
2827lor 70 . . . . . 6 ((a ∩ (a4 (b4 a)) ) ∪ ((a4 (b4 a)) ∩ (a4 (b4 a)) )) = ((a ∩ (a4 (b4 a)) ) ∪ 0)
29 or0 102 . . . . . 6 ((a ∩ (a4 (b4 a)) ) ∪ 0) = (a ∩ (a4 (b4 a)) )
3028, 29ax-r2 36 . . . . 5 ((a ∩ (a4 (b4 a)) ) ∪ ((a4 (b4 a)) ∩ (a4 (b4 a)) )) = (a ∩ (a4 (b4 a)) )
3125, 30ax-r2 36 . . . 4 ((a ∪ (a4 (b4 a))) ∩ (a4 (b4 a)) ) = (a ∩ (a4 (b4 a)) )
3221, 312or 72 . . 3 (((a ∩ (a4 (b4 a))) ∪ (a ∩ (a4 (b4 a)))) ∪ ((a ∪ (a4 (b4 a))) ∩ (a4 (b4 a)) )) = ((a4 (b4 a)) ∪ (a ∩ (a4 (b4 a)) ))
3310comcom2 183 . . . . . 6 (a4 (b4 a)) C a
3423comcom2 183 . . . . . 6 (a4 (b4 a)) C (a4 (b4 a))
3533, 34fh3 471 . . . . 5 ((a4 (b4 a)) ∪ (a ∩ (a4 (b4 a)) )) = (((a4 (b4 a)) ∪ a ) ∩ ((a4 (b4 a)) ∪ (a4 (b4 a)) ))
36 df-t 41 . . . . . . . 8 1 = ((a4 (b4 a)) ∪ (a4 (b4 a)) )
3736ax-r1 35 . . . . . . 7 ((a4 (b4 a)) ∪ (a4 (b4 a)) ) = 1
3837lan 77 . . . . . 6 (((a4 (b4 a)) ∪ a ) ∩ ((a4 (b4 a)) ∪ (a4 (b4 a)) )) = (((a4 (b4 a)) ∪ a ) ∩ 1)
39 an1 106 . . . . . 6 (((a4 (b4 a)) ∪ a ) ∩ 1) = ((a4 (b4 a)) ∪ a )
4038, 39ax-r2 36 . . . . 5 (((a4 (b4 a)) ∪ a ) ∩ ((a4 (b4 a)) ∪ (a4 (b4 a)) )) = ((a4 (b4 a)) ∪ a )
4135, 40ax-r2 36 . . . 4 ((a4 (b4 a)) ∪ (a ∩ (a4 (b4 a)) )) = ((a4 (b4 a)) ∪ a )
422ax-r5 38 . . . . 5 ((a4 (b4 a)) ∪ a ) = ((a ∪ ((ab) ∪ (ab ))) ∪ a )
43 or32 82 . . . . . 6 ((a ∪ ((ab) ∪ (ab ))) ∪ a ) = ((aa ) ∪ ((ab) ∪ (ab )))
44 oridm 110 . . . . . . . 8 (aa ) = a
4544ax-r5 38 . . . . . . 7 ((aa ) ∪ ((ab) ∪ (ab ))) = (a ∪ ((ab) ∪ (ab )))
462ax-r1 35 . . . . . . 7 (a ∪ ((ab) ∪ (ab ))) = (a4 (b4 a))
4745, 46ax-r2 36 . . . . . 6 ((aa ) ∪ ((ab) ∪ (ab ))) = (a4 (b4 a))
4843, 47ax-r2 36 . . . . 5 ((a ∪ ((ab) ∪ (ab ))) ∪ a ) = (a4 (b4 a))
4942, 48ax-r2 36 . . . 4 ((a4 (b4 a)) ∪ a ) = (a4 (b4 a))
5041, 49ax-r2 36 . . 3 ((a4 (b4 a)) ∪ (a ∩ (a4 (b4 a)) )) = (a4 (b4 a))
5132, 50ax-r2 36 . 2 (((a ∩ (a4 (b4 a))) ∪ (a ∩ (a4 (b4 a)))) ∪ ((a ∪ (a4 (b4 a))) ∩ (a4 (b4 a)) )) = (a4 (b4 a))
521, 51ax-r2 36 1 (a4 (a4 (b4 a))) = (a4 (b4 a))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ 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: (None)
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