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Theorem wdwom 1104
 Description: Prove 2-variable WOML rule in WDOL. This will make all WOML theorems available to us. The proof does not use ax-r3 439 or ax-wom 361. Since this is the same as ax-wom 361, from here on we will freely use those theorems invoking ax-wom 361.
Hypothesis
Ref Expression
wdwom.1 (a ∪ (ab)) = 1
Assertion
Ref Expression
wdwom (b ∪ (ab )) = 1

Proof of Theorem wdwom
StepHypRef Expression
1 df-i2 45 . . 3 (a2 b) = (b ∪ (ab ))
21ax-r1 35 . 2 (b ∪ (ab )) = (a2 b)
3 le1 146 . . 3 (a2 b) ≤ 1
4 df-i5 48 . . . . . 6 (a5 b) = (((ab) ∪ (ab)) ∪ (ab ))
5 df-i1 44 . . . . . . . . 9 (a1 b) = (a ∪ (ab))
6 wdwom.1 . . . . . . . . 9 (a ∪ (ab)) = 1
75, 6ax-r2 36 . . . . . . . 8 (a1 b) = 1
87wql1lem 287 . . . . . . 7 (ab) = 1
9 or4 84 . . . . . . . . . 10 (((ab) ∪ (ab)) ∪ ((ab) ∪ (ab ))) = (((ab) ∪ (ab) ) ∪ ((ab) ∪ (ab )))
10 anor1 88 . . . . . . . . . . . . 13 (ab ) = (ab)
1110ax-r1 35 . . . . . . . . . . . 12 (ab) = (ab )
1211lor 70 . . . . . . . . . . 11 ((ab) ∪ (ab) ) = ((ab) ∪ (ab ))
1312ax-r5 38 . . . . . . . . . 10 (((ab) ∪ (ab) ) ∪ ((ab) ∪ (ab ))) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
149, 13ax-r2 36 . . . . . . . . 9 (((ab) ∪ (ab)) ∪ ((ab) ∪ (ab ))) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
15 or12 80 . . . . . . . . 9 ((ab) ∪ (((ab) ∪ (ab)) ∪ (ab ))) = (((ab) ∪ (ab)) ∪ ((ab) ∪ (ab )))
16 df-cmtr 134 . . . . . . . . 9 C (a, b) = (((ab) ∪ (ab )) ∪ ((ab) ∪ (ab )))
1714, 15, 163tr1 63 . . . . . . . 8 ((ab) ∪ (((ab) ∪ (ab)) ∪ (ab ))) = C (a, b)
18 wdcom 1103 . . . . . . . 8 C (a, b) = 1
1917, 18ax-r2 36 . . . . . . 7 ((ab) ∪ (((ab) ∪ (ab)) ∪ (ab ))) = 1
208, 19skr0 242 . . . . . 6 (((ab) ∪ (ab)) ∪ (ab )) = 1
214, 20ax-r2 36 . . . . 5 (a5 b) = 1
2221ax-r1 35 . . . 4 1 = (a5 b)
23 i5lei2 348 . . . 4 (a5 b) ≤ (a2 b)
2422, 23bltr 138 . . 3 1 ≤ (a2 b)
253, 24lebi 145 . 2 (a2 b) = 1
262, 25ax-r2 36 1 (b ∪ (ab )) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12   →2 wi2 13   →5 wi5 16   C wcmtr 29 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-wdol 1102 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-i5 48  df-le1 130  df-le2 131  df-cmtr 134 This theorem is referenced by: (None)
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