Theorem wlbtr 398

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

Hypotheses

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

Assertion

Ref	Expression
wlbtr	(a ≤2 c) = 1

Proof of Theorem wlbtr

Step	Hyp	Ref	Expression
1		wlbtr.2	. . . . 5 (b ≡ c) = 1
2	1	wr1 197	. . . 4 (c ≡ b) = 1
3	2	wlan 370	. . 3 ((a ∩ c) ≡ (a ∩ b)) = 1
4		wlbtr.1	. . . 4 (a ≤2 b) = 1
5	4	wdf2le2 386	. . 3 ((a ∩ b) ≡ a) = 1
6	3, 5	wr2 371	. 2 ((a ∩ c) ≡ a) = 1
7	6	wdf2le1 385	1 (a ≤2 c) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∩ wa 7  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  ska4  433