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Theorem wql2lem4 291
 Description: Lemma for →2 WQL axiom.
Hypotheses
Ref Expression
wql2lem4.1 (((ab ) ∪ (ab)) →2 (a ∪ (ab))) = 1
wql2lem4.2 ((a1 b) ∪ (ab )) = 1
Assertion
Ref Expression
wql2lem4 (a1 b) = 1

Proof of Theorem wql2lem4
StepHypRef Expression
1 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
2 id 59 . 2 (a ∪ (ab)) = (a ∪ (ab))
3 ax-a2 31 . . . 4 ((ab ) ∪ (a ∪ (ab))) = ((a ∪ (ab)) ∪ (ab ))
41ax-r5 38 . . . . 5 ((a1 b) ∪ (ab )) = ((a ∪ (ab)) ∪ (ab ))
54ax-r1 35 . . . 4 ((a ∪ (ab)) ∪ (ab )) = ((a1 b) ∪ (ab ))
6 wql2lem4.2 . . . 4 ((a1 b) ∪ (ab )) = 1
73, 5, 63tr 65 . . 3 ((ab ) ∪ (a ∪ (ab))) = 1
8 wql2lem4.1 . . . 4 (((ab ) ∪ (ab)) →2 (a ∪ (ab))) = 1
98wql2lem2 289 . . 3 (((ab ) ∪ (a ∪ (ab))) ∪ (a ∪ (ab))) = 1
107, 9skr0 242 . 2 (a ∪ (ab)) = 1
111, 2, 103tr 65 1 (a1 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-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 This theorem is referenced by: (None)
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