Theorem ltsub23d

Description: 'Less than' relationship between subtraction and addition. (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 )
	ltsub23d.4	|- ( ph -> ( A - B ) < C )
Assertion	ltsub23d	|- ( 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		ltsub23d.4	|- ( ph -> ( A - B ) < C )
5		ltsub23	|- ( ( 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 )