Theorem i3i0tr 542

Description: Transitive inference. (Contributed by NM, 9-Nov-1997.)

Hypotheses

Ref	Expression
i3i0tr.1	(a →3 b) = 1
i3i0tr.2	(b →3 (b →3 c)) = 1

Assertion

Ref	Expression
i3i0tr	(a →3 (a →3 c)) = 1

Proof of Theorem i3i0tr

Step	Hyp	Ref	Expression
1		i3i0tr.2	. . . 4 (b →3 (b →3 c)) = 1
2	1	i3i0 513	. . 3 (b⊥ ∪ c) = 1
3		i3i0tr.1	. . . . 5 (a →3 b) = 1
4	3	binr1 517	. . . 4 (b⊥ →3 a⊥ ) = 1
5	4	i3ror 532	. . 3 ((b⊥ ∪ c) →3 (a⊥ ∪ c)) = 1
6	2, 5	skmp3 245	. 2 (a⊥ ∪ c) = 1
7	6	i0i3 512	1 (a →3 (a →3 c)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6  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: (None)