Theorem ledir 167

Description: Half of distributive law.

Assertion

Ref	Expression
ledir	((b ^ a) v (c ^ a)) =< ((b v c) ^ a)

Proof of Theorem ledir

Step	Hyp	Ref	Expression
1		ledi 166	. 2 ((a ^ b) v (a ^ c)) =< (a ^ (b v c))
2		ancom 68	. . 3 (b ^ a) = (a ^ b)
3		ancom 68	. . 3 (c ^ a) = (a ^ c)
4	2, 3	2or 67	. 2 ((b ^ a) v (c ^ a)) = ((a ^ b) v (a ^ c))
5		ancom 68	. 2 ((b v c) ^ a) = (a ^ (b v c))
6	1, 4, 5	le3tr1 132	1 ((b ^ a) v (c ^ a)) =< ((b v c) ^ a)

Syntax hints:   =< wle 2   v wo 6   ^ wa 7

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-le1 122  df-le2 123

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