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Theorem wql1 293
 Description: The 2nd hypothesis is the first →1 WQL axiom. We show it implies the WOM law.
Hypotheses
Ref Expression
wql1.1 (a1 b) = 1
wql1.2 ((ac) →1 (bc)) = 1
wql1.3 c = b
Assertion
Ref Expression
wql1 (a2 b) = 1

Proof of Theorem wql1
StepHypRef Expression
1 df-i2 45 . 2 (a2 b) = (b ∪ (ab ))
2 anor3 90 . . 3 (ab ) = (ab)
32lor 70 . 2 (b ∪ (ab )) = (b ∪ (ab) )
4 ax-a2 31 . . 3 (b ∪ (ab) ) = ((ab)b)
5 wql1.3 . . . . . . . . 9 c = b
65lor 70 . . . . . . . 8 (bc) = (bb)
7 oridm 110 . . . . . . . 8 (bb) = b
86, 7ax-r2 36 . . . . . . 7 (bc) = b
98ud1lem0a 255 . . . . . 6 ((ac) →1 (bc)) = ((ac) →1 b)
109ax-r1 35 . . . . 5 ((ac) →1 b) = ((ac) →1 (bc))
115lor 70 . . . . . 6 (ac) = (ab)
1211ud1lem0b 256 . . . . 5 ((ac) →1 b) = ((ab) →1 b)
13 wql1.2 . . . . 5 ((ac) →1 (bc)) = 1
1410, 12, 133tr2 64 . . . 4 ((ab) →1 b) = 1
1514wql1lem 287 . . 3 ((ab)b) = 1
164, 15ax-r2 36 . 2 (b ∪ (ab) ) = 1
171, 3, 163tr 65 1 (a2 b) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12   →2 wi2 13 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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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