Theorem i33tr2 530

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

Hypotheses

Ref	Expression
i33tr2.1	(a →3 b) = 1
i33tr2.2	a = c
i33tr2.3	b = d

Assertion

Ref	Expression
i33tr2	(c →3 d) = 1

Proof of Theorem i33tr2

Step	Hyp	Ref	Expression
1		i33tr2.1	. 2 (a →3 b) = 1
2		i33tr2.2	. . 3 a = c
3	2	ax-r1 35	. 2 c = a
4		i33tr2.3	. . 3 b = d
5	4	ax-r1 35	. 2 d = b
6	1, 3, 5	i33tr1 529	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: (None)