Theorem bii3 498

Description: Biconditional implies Kalmbach implication.

Assertion

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

Proof of Theorem bii3

Step	Hyp	Ref	Expression
1		i3bi 478	. . . 4 ((a →3 b) ∩ (b →3 a)) = (a ≡ b)
2	1	ax-r1 34	. . 3 (a ≡ b) = ((a →3 b) ∩ (b →3 a))
3		lea 152	. . 3 ((a →3 b) ∩ (b →3 a)) ≤ (a →3 b)
4	2, 3	bltr 130	. 2 (a ≡ b) ≤ (a →3 b)
5	4	lei3 238	1 ((a ≡ b) →3 (a →3 b)) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∩ wa 7  1wt 9   →₃ wi3 15

This theorem is referenced by:  i3th6 530

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org