Theorem oa4lem3 919

Description: Lemma for 3-var to 4-var OA.

Hypotheses

Ref	Expression
oa4lem1.1	a =< b_|_
oa4lem1.2	c =< d_|_

Assertion

Ref	Expression
oa4lem3	((a v b) ^ (c v d)) =< ((b v d)_|_ v (((a v c)_|_ ->2 b) ^ ((a v c)_|_ ->2 d)))

Proof of Theorem oa4lem3

Step	Hyp	Ref	Expression
1		oa4lem1.1	. . . 4 a =< b_|_
2		oa4lem1.2	. . . 4 c =< d_|_
3	1, 2	oa4lem1 917	. . 3 (a v b) =< ((a v c)_|_ ->2 b)
4	1, 2	oa4lem2 918	. . 3 (c v d) =< ((a v c)_|_ ->2 d)
5	3, 4	le2an 161	. 2 ((a v b) ^ (c v d)) =< (((a v c)_|_ ->2 b) ^ ((a v c)_|_ ->2 d))
6		leor 151	. 2 (((a v c)_|_ ->2 b) ^ ((a v c)_|_ ->2 d)) =< ((b v d)_|_ v (((a v c)_|_ ->2 b) ^ ((a v c)_|_ ->2 d)))
7	5, 6	letr 129	1 ((a v b) ^ (c v d)) =< ((b v d)_|_ v (((a v c)_|_ ->2 b) ^ ((a v c)_|_ ->2 d)))

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->2 wi2 14

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i2 44  df-le1 122  df-le2 123

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