Theorem wlbtr 380

Description: Transitive inference.

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 189	. . . 4 (c ≡ b) = 1
3	2	wlan 352	. . 3 ((a ∩ c) ≡ (a ∩ b)) = 1
4		wlbtr.1	. . . 4 (a ≤2 b) = 1
5	4	wdf2le2 368	. . 3 ((a ∩ b) ≡ a) = 1
6	3, 5	wr2 353	. 2 ((a ∩ c) ≡ a) = 1
7	6	wdf2le1 367	1 (a ≤2 c) = 1

Syntax hints:   = wb 1   ≡ tb 5   ∩ wa 7  1wt 9   ≤₂ wle2 11

This theorem is referenced by:  wle3tr1 381  wledi 387  wledio 388  ska4 415

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-le 121  df-le1 122  df-le2 123

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