Theorem wletr 378

Description: Transitive law for l.e.

Hypotheses

Ref	Expression
wletr.1	(a =<2 b) = 1
wletr.2	(b =<2 c) = 1

Assertion

Ref	Expression
wletr	(a =<2 c) = 1

Proof of Theorem wletr

Step	Hyp	Ref	Expression
1		wletr.1	. . . . . . . 8 (a =<2 b) = 1
2	1	wdf-le2 361	. . . . . . 7 ((a v b) == b) = 1
3	2	wr5-2v 348	. . . . . 6 (((a v b) v c) == (b v c)) = 1
4	3	wr1 189	. . . . 5 ((b v c) == ((a v b) v c)) = 1
5		wletr.2	. . . . . 6 (b =<2 c) = 1
6	5	wdf-le2 361	. . . . 5 ((b v c) == c) = 1
7		ax-a3 31	. . . . . 6 ((a v b) v c) = (a v (b v c))
8	7	bi1 110	. . . . 5 (((a v b) v c) == (a v (b v c))) = 1
9	4, 6, 8	w3tr2 357	. . . 4 (c == (a v (b v c))) = 1
10	9	wlan 352	. . 3 ((a ^ c) == (a ^ (a v (b v c)))) = 1
11		a5c 113	. . . 4 (a ^ (a v (b v c))) = a
12	11	bi1 110	. . 3 ((a ^ (a v (b v c))) == a) = 1
13	10, 12	wr2 353	. 2 ((a ^ c) == a) = 1
14	13	wdf2le1 367	1 (a =<2 c) = 1

Syntax hints:   = wb 1   v wo 6   ^ wa 7  1wt 9   =<2 wle2 11

This theorem is referenced by:  wle2or 385  wle2an 386  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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