Theorem i0i3tr 523

Description: Transitive inference.

Hypotheses

Ref	Expression
i0i3tr.1	(a →3 (a →3 b)) = 1
i0i3tr.2	(b →3 c) = 1

Assertion

Ref	Expression
i0i3tr	(a →3 (a →3 c)) = 1

Proof of Theorem i0i3tr

Step	Hyp	Ref	Expression
1		i0i3tr.1	. . . 4 (a →3 (a →3 b)) = 1
2	1	i3i0 495	. . 3 (a⊥ ∪ b) = 1
3		i0i3tr.2	. . . 4 (b →3 c) = 1
4	3	i3lor 515	. . 3 ((a⊥ ∪ b) →3 (a⊥ ∪ c)) = 1
5	2, 4	skmp3 237	. 2 (a⊥ ∪ c) = 1
6	5	i0i3 494	1 (a →3 (a →3 c)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6  1wt 9   →₃ wi3 15

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