Theorem wbltr 397

Description: Transitive inference. (Contributed by NM, 13-Oct-1997.)

Hypotheses

Ref	Expression
wbltr.1	(a ≡ b) = 1
wbltr.2	(b ≤2 c) = 1

Assertion

Ref	Expression
wbltr	(a ≤2 c) = 1

Proof of Theorem wbltr

Step	Hyp	Ref	Expression
1		wbltr.1	. . . 4 (a ≡ b) = 1
2	1	wr5-2v 366	. . 3 ((a ∪ c) ≡ (b ∪ c)) = 1
3		wbltr.2	. . . 4 (b ≤2 c) = 1
4	3	wdf-le2 379	. . 3 ((b ∪ c) ≡ c) = 1
5	2, 4	wr2 371	. 2 ((a ∪ c) ≡ c) = 1
6	5	wdf-le1 378	1 (a ≤2 c) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∪ wo 6  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:  wle3tr1  399  wledi  405  wledio  406  wlem14  430