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Theorem 1b 117
 Description: Identity law.
Assertion
Ref Expression
1b (1 ≡ a) = a

Proof of Theorem 1b
StepHypRef Expression
1 dfb 94 . 2 (1 ≡ a) = ((1 ∩ a) ∪ (1a ))
2 ancom 74 . . . . 5 (1 ∩ a) = (a ∩ 1)
3 ancom 74 . . . . . 6 (1a ) = (a ∩ 1 )
4 df-f 42 . . . . . . . 8 0 = 1
54ax-r1 35 . . . . . . 7 1 = 0
65lan 77 . . . . . 6 (a ∩ 1 ) = (a ∩ 0)
73, 6ax-r2 36 . . . . 5 (1a ) = (a ∩ 0)
82, 72or 72 . . . 4 ((1 ∩ a) ∪ (1a )) = ((a ∩ 1) ∪ (a ∩ 0))
9 an1 106 . . . . 5 (a ∩ 1) = a
10 an0 108 . . . . 5 (a ∩ 0) = 0
119, 102or 72 . . . 4 ((a ∩ 1) ∪ (a ∩ 0)) = (a ∪ 0)
128, 11ax-r2 36 . . 3 ((1 ∩ a) ∪ (1a )) = (a ∪ 0)
13 or0 102 . . 3 (a ∪ 0) = a
1412, 13ax-r2 36 . 2 ((1 ∩ a) ∪ (1a )) = a
151, 14ax-r2 36 1 (1 ≡ a) = a
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42 This theorem is referenced by:  wr3  198  woml6  436  woml7  437  r3b  442
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