Theorem i2id 268

Description: Identity law for Dishkant conditional.

Assertion

Ref	Expression
i2id	(a →2 a) = 1

Proof of Theorem i2id

Step	Hyp	Ref	Expression
1		df-i2 44	. 2 (a →2 a) = (a ∪ (a⊥ ∩ a⊥ ))
2		anidm 103	. . . 4 (a⊥ ∩ a⊥ ) = a⊥
3	2	lor 66	. . 3 (a ∪ (a⊥ ∩ a⊥ )) = (a ∪ a⊥ )
4		df-t 40	. . . 4 1 = (a ∪ a⊥ )
5	4	ax-r1 34	. . 3 (a ∪ a⊥ ) = 1
6	3, 5	ax-r2 35	. 2 (a ∪ (a⊥ ∩ a⊥ )) = 1
7	1, 6	ax-r2 35	1 (a →2 a) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₂ wi2 14

This theorem is referenced by:  oago3.29 871

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i2 44

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