Theorem wle3tr1 381

Description: Transitive inference useful for introducing definitions.

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 379	. 2 (c =<2 b) = 1
4		wle3tr1.3	. . 3 (d == b) = 1
5	4	wr1 189	. 2 (b == d) = 1
6	3, 5	wlbtr 380	1 (c =<2 d) = 1

Syntax hints:   = wb 1   == tb 5  1wt 9   =<2 wle2 11

This theorem is referenced by:  wle3tr2 382  wle2or 385  wle2an 386

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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