Theorem w3tr1 374

Description: Transitive inference useful for introducing definitions. (Contributed by NM, 13-Oct-1997.)

Hypotheses

Ref	Expression
w3tr1.1	(a ≡ b) = 1
w3tr1.2	(c ≡ a) = 1
w3tr1.3	(d ≡ b) = 1

Assertion

Ref	Expression
w3tr1	(c ≡ d) = 1

Proof of Theorem w3tr1

Step	Hyp	Ref	Expression
1		w3tr1.2	. 2 (c ≡ a) = 1
2		w3tr1.1	. . 3 (a ≡ b) = 1
3		w3tr1.3	. . . 4 (d ≡ b) = 1
4	3	wr1 197	. . 3 (b ≡ d) = 1
5	2, 4	wr2 371	. 2 (a ≡ d) = 1
6	1, 5	wr2 371	1 (c ≡ d) = 1

Syntax hints:   = wb 1   ≡ tb 5  1wt 8

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-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  w3tr2  375  wcomlem  382  wbctr  410  wcomcom5  420  wfh1  423  wfh2  424  wdid0id5  1111  wdid0id1  1112  wdid0id2  1113  wdid0id3  1114