Theorem i2id 276

Description: Identity law for Dishkant conditional. (Contributed by NM, 26-Jun-2003.)

Assertion

Ref	Expression
i2id	(a →2 a) = 1

Proof of Theorem i2id

Step	Hyp	Ref	Expression
1		df-i2 45	. 2 (a →2 a) = (a ∪ (a⊥ ∩ a⊥ ))
2		anidm 111	. . . 4 (a⊥ ∩ a⊥ ) = a⊥
3	2	lor 70	. . 3 (a ∪ (a⊥ ∩ a⊥ )) = (a ∪ a⊥ )
4		df-t 41	. . . 4 1 = (a ∪ a⊥ )
5	4	ax-r1 35	. . 3 (a ∪ a⊥ ) = 1
6	3, 5	ax-r2 36	. 2 (a ∪ (a⊥ ∩ a⊥ )) = 1
7	1, 6	ax-r2 36	1 (a →2 a) = 1

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-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

This theorem is referenced by:  oago3.29  889