Theorem w3tr2 375

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

Hypotheses

Ref	Expression
w3tr2.1	(a ≡ b) = 1
w3tr2.2	(a ≡ c) = 1
w3tr2.3	(b ≡ d) = 1

Assertion

Ref	Expression
w3tr2	(c ≡ d) = 1

Proof of Theorem w3tr2

Step	Hyp	Ref	Expression
1		w3tr2.1	. 2 (a ≡ b) = 1
2		w3tr2.2	. . 3 (a ≡ c) = 1
3	2	wr1 197	. 2 (c ≡ a) = 1
4		w3tr2.3	. . 3 (b ≡ d) = 1
5	4	wr1 197	. 2 (d ≡ b) = 1
6	1, 3, 5	w3tr1 374	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:  wom4  380  wom5  381  wcomlem  382  wlecon  395  wletr  396  wcom3i  422  wfh3  425  wfh4  426  wlem14  430