Theorem i33tr1 529

Description: Transitive inference useful for introducing definitions. (Contributed by NM, 7-Nov-1997.)

Hypotheses

Ref	Expression
i33tr1.1	(a →3 b) = 1
i33tr1.2	c = a
i33tr1.3	d = b

Assertion

Ref	Expression
i33tr1	(c →3 d) = 1

Proof of Theorem i33tr1

Step	Hyp	Ref	Expression
1		i33tr1.2	. . 3 c = a
2		i33tr1.1	. . 3 (a →3 b) = 1
3	1, 2	bi3tr 527	. 2 (c →3 b) = 1
4		i33tr1.3	. . 3 d = b
5	4	ax-r1 35	. 2 b = d
6	3, 5	i3btr 528	1 (c →3 d) = 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:  i33tr2  530  i3con1  531  i3ran  535  i3lan  536