Theorem bii3 516

Description: Biconditional implies Kalmbach implication. (Contributed by NM, 9-Nov-1997.)

Assertion

Ref	Expression
bii3	((a ≡ b) →3 (a →3 b)) = 1

Proof of Theorem bii3

Step	Hyp	Ref	Expression
1		i3bi 496	. . . 4 ((a →3 b) ∩ (b →3 a)) = (a ≡ b)
2	1	ax-r1 35	. . 3 (a ≡ b) = ((a →3 b) ∩ (b →3 a))
3		lea 160	. . 3 ((a →3 b) ∩ (b →3 a)) ≤ (a →3 b)
4	2, 3	bltr 138	. 2 (a ≡ b) ≤ (a →3 b)
5	4	lei3 246	1 ((a ≡ b) →3 (a →3 b)) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∩ wa 7  1wt 8   →₃ wi3 14

This theorem was proved from 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 theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  i3th6  548