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Theorem 1i1 274
 Description: Antecedent of 1 on Sasaki conditional.
Assertion
Ref Expression
1i1 (1 →1 a) = a

Proof of Theorem 1i1
StepHypRef Expression
1 df-i1 44 . 2 (1 →1 a) = (1 ∪ (1 ∩ a))
2 df-f 42 . . . . 5 0 = 1
32ax-r1 35 . . . 4 1 = 0
4 ancom 74 . . . . 5 (1 ∩ a) = (a ∩ 1)
5 an1 106 . . . . 5 (a ∩ 1) = a
64, 5ax-r2 36 . . . 4 (1 ∩ a) = a
73, 62or 72 . . 3 (1 ∪ (1 ∩ a)) = (0 ∪ a)
8 ax-a2 31 . . . 4 (0 ∪ a) = (a ∪ 0)
9 or0 102 . . . 4 (a ∪ 0) = a
108, 9ax-r2 36 . . 3 (0 ∪ a) = a
117, 10ax-r2 36 . 2 (1 ∪ (1 ∩ a)) = a
121, 11ax-r2 36 1 (1 →1 a) = a
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →1 wi1 12 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  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 This theorem is referenced by:  oa3-6lem  980
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