Theorem i1id 275

Description: Identity law for Sasaki conditional. (Contributed by NM, 25-Dec-1998.)

Assertion

Ref	Expression
i1id	(a →1 a) = 1

Proof of Theorem i1id

Step	Hyp	Ref	Expression
1		df-i1 44	. 2 (a →1 a) = (a⊥ ∪ (a ∩ a))
2		ax-a2 31	. . 3 (a⊥ ∪ a) = (a ∪ a⊥ )
3		anidm 111	. . . 4 (a ∩ a) = a
4	3	lor 70	. . 3 (a⊥ ∪ (a ∩ a)) = (a⊥ ∪ a)
5		df-t 41	. . 3 1 = (a ∪ a⊥ )
6	2, 4, 5	3tr1 63	. 2 (a⊥ ∪ (a ∩ a)) = 1
7	1, 6	ax-r2 36	1 (a →1 a) = 1

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44

This theorem is used by:  oa3-2lemb  979