Theorem subled

Description: Swap subtrahends in an inequality. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
	ltadd1d.3	|- ( ph -> C e. RR )
	subled.4	|- ( ph -> ( A - B ) <_ C )
Assertion	subled	|- ( ph -> ( A - C ) <_ B )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	|- ( ph -> A e. RR )
2		ltnegd.2	|- ( ph -> B e. RR )
3		ltadd1d.3	|- ( ph -> C e. RR )
4		subled.4	|- ( ph -> ( A - B ) <_ C )
5		suble	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) <_ C <-> ( A - C ) <_ B ) )
6	1 2 3 5	syl3anc	|- ( ph -> ( ( A - B ) <_ C <-> ( A - C ) <_ B ) )
7	4 6	mpbid	|- ( ph -> ( A - C ) <_ B )