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Theorem wql2lem3 290
 Description: Lemma for →2 WQL axiom.
Hypothesis
Ref Expression
wql2lem3.1 (a2 b) = 1
Assertion
Ref Expression
wql2lem3 ((ab ) →2 a ) = 1

Proof of Theorem wql2lem3
StepHypRef Expression
1 df-i2 45 . 2 ((ab ) →2 a ) = (a ∪ ((ab )a ))
2 oran2 92 . . . . . 6 (ab) = (ab )
32ax-r1 35 . . . . 5 (ab ) = (ab)
43ran 78 . . . 4 ((ab )a ) = ((ab) ∩ a )
5 ancom 74 . . . 4 ((ab) ∩ a ) = (a ∩ (ab))
64, 5ax-r2 36 . . 3 ((ab )a ) = (a ∩ (ab))
76lor 70 . 2 (a ∪ ((ab )a )) = (a ∪ (a ∩ (ab)))
8 wql2lem3.1 . . . 4 (a2 b) = 1
98wql2lem 288 . . 3 (ab) = 1
10 omlem2 128 . . 3 ((ab) ∪ (a ∪ (a ∩ (ab)))) = 1
119, 10skr0 242 . 2 (a ∪ (a ∩ (ab))) = 1
121, 7, 113tr 65 1 ((ab ) →2 a ) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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-i2 45  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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