Theorem int-ineqmvtd

Description: IneqMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)

	Ref	Expression
Hypotheses	int-ineqmvtd.1	$$⊢\phi \to B\in ℝ$$
	int-ineqmvtd.2	$$⊢\phi \to C\in ℝ$$
	int-ineqmvtd.3	$$⊢\phi \to D\in ℝ$$
	int-ineqmvtd.4	$$⊢\phi \to B\le A$$
	int-ineqmvtd.5	$$⊢\phi \to A=C+D$$
Assertion	int-ineqmvtd	$$⊢\phi \to B-D\le C$$

Proof

Step	Hyp	Ref	Expression
1		int-ineqmvtd.1	$$⊢\phi \to B\in ℝ$$
2		int-ineqmvtd.2	$$⊢\phi \to C\in ℝ$$
3		int-ineqmvtd.3	$$⊢\phi \to D\in ℝ$$
4		int-ineqmvtd.4	$$⊢\phi \to B\le A$$
5		int-ineqmvtd.5	$$⊢\phi \to A=C+D$$
6	4 5	breqtrd	$$⊢\phi \to B\le C+D$$
7	1 3 2	lesubaddd	$$⊢\phi \to \left(B-D\le C\leftrightarrow B\le C+D\right)$$
8	6 7	mpbird	$$⊢\phi \to B-D\le C$$