Theorem bi3tr 527

Description: Transitive inference. (Contributed by NM, 7-Nov-1997.)

Hypotheses

Ref	Expression
bi3tr.1	a = b
bi3tr.2	(b →3 c) = 1

Assertion

Ref	Expression
bi3tr	(a →3 c) = 1

Proof of Theorem bi3tr

Step	Hyp	Ref	Expression
1		bi3tr.2	. 2 (b →3 c) = 1
2		bi3tr.1	. . . 4 a = b
3	2	ri3 253	. . 3 (a →3 c) = (b →3 c)
4	3	bi1 118	. 2 ((a →3 c) ≡ (b →3 c)) = 1
5	1, 4	wwbmpr 206	1 (a →3 c) = 1

Syntax hints:   = wb 1  1wt 8   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46

This theorem is referenced by:  i33tr1  529