Theorem w3tr2 357

Description: Transitive inference useful for eliminating definitions.

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 189	. 2 (c ≡ a) = 1
4		w3tr2.3	. . 3 (b ≡ d) = 1
5	4	wr1 189	. 2 (d ≡ b) = 1
6	1, 3, 5	w3tr1 356	1 (c ≡ d) = 1

Syntax hints:   = wb 1   ≡ tb 5  1wt 9

This theorem is referenced by:  wom4 362  wom5 363  wcomlem 364  wlecon 377  wletr 378  wcom3i 404  wfh3 407  wfh4 408  wlem14 412

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-wom 343

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i1 43  df-i2 44  df-le1 122  df-le2 123

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