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Theorem i3or 497
 Description: Kalmbach implication OR builder. (Contributed by NM, 26-Dec-1997.)
Assertion
Ref Expression
i3or ((ab) ∪ ((ac) →3 (bc))) = 1

Proof of Theorem i3or
StepHypRef Expression
1 le1 146 . 2 ((ab) ∪ ((ac) →3 (bc))) ≤ 1
2 ka4ot 435 . . . 4 ((ab) ∪ ((ac) ≡ (bc))) = 1
32ax-r1 35 . . 3 1 = ((ab) ∪ ((ac) ≡ (bc)))
4 i3bi 496 . . . . . 6 (((ac) →3 (bc)) ∩ ((bc) →3 (ac))) = ((ac) ≡ (bc))
54ax-r1 35 . . . . 5 ((ac) ≡ (bc)) = (((ac) →3 (bc)) ∩ ((bc) →3 (ac)))
6 lea 160 . . . . 5 (((ac) →3 (bc)) ∩ ((bc) →3 (ac))) ≤ ((ac) →3 (bc))
75, 6bltr 138 . . . 4 ((ac) ≡ (bc)) ≤ ((ac) →3 (bc))
87lelor 166 . . 3 ((ab) ∪ ((ac) ≡ (bc))) ≤ ((ab) ∪ ((ac) →3 (bc)))
93, 8bltr 138 . 2 1 ≤ ((ab) ∪ ((ac) →3 (bc)))
101, 9lebi 145 1 ((ab) ∪ ((ac) →3 (bc))) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   →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-wom 361  ax-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-i3 46  df-le 129  df-le1 130  df-le2 131  df-c1 132  df-c2 133  df-cmtr 134 This theorem is referenced by: (None)
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