Theorem wle3tr1 399

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

Hypotheses

Ref	Expression
wle3tr1.1	(a ≤2 b) = 1
wle3tr1.2	(c ≡ a) = 1
wle3tr1.3	(d ≡ b) = 1

Assertion

Ref	Expression
wle3tr1	(c ≤2 d) = 1

Proof of Theorem wle3tr1

Step	Hyp	Ref	Expression
1		wle3tr1.2	. . 3 (c ≡ a) = 1
2		wle3tr1.1	. . 3 (a ≤2 b) = 1
3	1, 2	wbltr 397	. 2 (c ≤2 b) = 1
4		wle3tr1.3	. . 3 (d ≡ b) = 1
5	4	wr1 197	. 2 (b ≡ d) = 1
6	3, 5	wlbtr 398	1 (c ≤2 d) = 1

Syntax hints:   = wb 1   ≡ tb 5  1wt 8   ≤₂ wle2 10

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-le 129  df-le1 130  df-le2 131

This theorem is referenced by:  wle3tr2  400  wle2or  403  wle2an  404