Theorem i33tr1 511

Description: Transitive inference useful for introducing definitions.

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 509	. 2 (c →3 b) = 1
4		i33tr1.3	. . 3 d = b
5	4	ax-r1 34	. 2 b = d
6	3, 5	i3btr 510	1 (c →3 d) = 1

Syntax hints:   = wb 1  1wt 9   →₃ wi3 15

This theorem is referenced by:  i33tr2 512  i3con1 513  i3ran 517  i3lan 518

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i3 45

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