Theorem w3tr1 356

Description: Transitive inference useful for introducing definitions.

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 189	. . 3 (b ≡ d) = 1
5	2, 4	wr2 353	. 2 (a ≡ d) = 1
6	1, 5	wr2 353	1 (c ≡ d) = 1

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

This theorem is referenced by:  w3tr2 357  wcomlem 364  wbctr 392  wcomcom5 402  wfh1 405  wfh2 406

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

metamath.org