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Theorem i3orlem7 558
 Description: Lemma for Kalmbach implication OR builder.
Assertion
Ref Expression
i3orlem7 (ab ) ≤ ((a3 b) ∪ ((ac) →3 (bc)))

Proof of Theorem i3orlem7
StepHypRef Expression
1 lea 160 . . . . . . 7 (ab ) ≤ a
2 leo 158 . . . . . . 7 a ≤ (ab)
31, 2letr 137 . . . . . 6 (ab ) ≤ (ab)
4 leo 158 . . . . . 6 (ab ) ≤ ((ab ) ∪ (ab))
53, 4ler2an 173 . . . . 5 (ab ) ≤ ((ab) ∩ ((ab ) ∪ (ab)))
65ler 149 . . . 4 (ab ) ≤ (((ab) ∩ ((ab ) ∪ (ab))) ∪ ((ab) ∩ (a ∩ (ab ))))
7 i3n1 249 . . . . . . 7 (a3 b ) = (((ab ) ∪ (ab)) ∪ (a ∩ (ab )))
87lan 77 . . . . . 6 ((ab) ∩ (a3 b )) = ((ab) ∩ (((ab ) ∪ (ab)) ∪ (a ∩ (ab ))))
9 comor1 461 . . . . . . . . 9 (ab) C a
10 comor2 462 . . . . . . . . . 10 (ab) C b
1110comcom2 183 . . . . . . . . 9 (ab) C b
129, 11com2an 484 . . . . . . . 8 (ab) C (ab )
139, 10com2an 484 . . . . . . . 8 (ab) C (ab)
1412, 13com2or 483 . . . . . . 7 (ab) C ((ab ) ∪ (ab))
159comcom2 183 . . . . . . . 8 (ab) C a
169, 11com2or 483 . . . . . . . 8 (ab) C (ab )
1715, 16com2an 484 . . . . . . 7 (ab) C (a ∩ (ab ))
1814, 17fh1 469 . . . . . 6 ((ab) ∩ (((ab ) ∪ (ab)) ∪ (a ∩ (ab )))) = (((ab) ∩ ((ab ) ∪ (ab))) ∪ ((ab) ∩ (a ∩ (ab ))))
198, 18ax-r2 36 . . . . 5 ((ab) ∩ (a3 b )) = (((ab) ∩ ((ab ) ∪ (ab))) ∪ ((ab) ∩ (a ∩ (ab ))))
2019ax-r1 35 . . . 4 (((ab) ∩ ((ab ) ∪ (ab))) ∪ ((ab) ∩ (a ∩ (ab )))) = ((ab) ∩ (a3 b ))
216, 20lbtr 139 . . 3 (ab ) ≤ ((ab) ∩ (a3 b ))
2221ler 149 . 2 (ab ) ≤ (((ab) ∩ (a3 b )) ∪ ((ac) →3 (bc)))
23 i3orlem6 557 . . 3 ((a3 b) ∪ ((ac) →3 (bc))) = (((ab) ∩ (a3 b )) ∪ ((ac) →3 (bc)))
2423ax-r1 35 . 2 (((ab) ∩ (a3 b )) ∪ ((ac) →3 (bc))) = ((a3 b) ∪ ((ac) →3 (bc)))
2522, 24lbtr 139 1 (ab ) ≤ ((a3 b) ∪ ((ac) →3 (bc)))
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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