Theorem i2bi 722

Description: Dishkant implication expressed with biconditional. (Contributed by NM, 20-Nov-1998.)

Assertion

Ref	Expression
i2bi	(a →2 b) = (b ∪ (a ≡ b))

Proof of Theorem i2bi

Step	Hyp	Ref	Expression
1		leor 159	. . . 4 (a⊥ ∩ b⊥ ) ≤ ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
2	1	lelor 166	. . 3 (b ∪ (a⊥ ∩ b⊥ )) ≤ (b ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
3		df-i2 45	. . 3 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
4		dfb 94	. . . 4 (a ≡ b) = ((a ∩ b) ∪ (a⊥ ∩ b⊥ ))
5	4	lor 70	. . 3 (b ∪ (a ≡ b)) = (b ∪ ((a ∩ b) ∪ (a⊥ ∩ b⊥ )))
6	2, 3, 5	le3tr1 140	. 2 (a →2 b) ≤ (b ∪ (a ≡ b))
7		leo 158	. . . 4 b ≤ (b ∪ (a⊥ ∩ b⊥ ))
8	3	ax-r1 35	. . . 4 (b ∪ (a⊥ ∩ b⊥ )) = (a →2 b)
9	7, 8	lbtr 139	. . 3 b ≤ (a →2 b)
10		u2lembi 721	. . . . 5 ((a →2 b) ∩ (b →2 a)) = (a ≡ b)
11	10	ax-r1 35	. . . 4 (a ≡ b) = ((a →2 b) ∩ (b →2 a))
12		lea 160	. . . 4 ((a →2 b) ∩ (b →2 a)) ≤ (a →2 b)
13	11, 12	bltr 138	. . 3 (a ≡ b) ≤ (a →2 b)
14	9, 13	lel2or 170	. 2 (b ∪ (a ≡ b)) ≤ (a →2 b)
15	6, 14	lebi 145	1 (a →2 b) = (b ∪ (a ≡ b))

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is used by:  mloa  1018