Theorem resubaddd

Description: Relationship between subtraction and addition. Based on subaddd . (Contributed by Steven Nguyen, 8-Jan-2023)

	Ref	Expression
Hypotheses	resubaddd.1	$$⊢\phi \to A\in ℝ$$
	resubaddd.2	$$⊢\phi \to B\in ℝ$$
	resubaddd.3	$$⊢\phi \to C\in ℝ$$
Assertion	resubaddd	$$⊢\phi \to \left(A{-}_{ℝ}B=C\leftrightarrow B+C=A\right)$$

Proof

Step	Hyp	Ref	Expression
1		resubaddd.1	$$⊢\phi \to A\in ℝ$$
2		resubaddd.2	$$⊢\phi \to B\in ℝ$$
3		resubaddd.3	$$⊢\phi \to C\in ℝ$$
4		resubadd	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge C\in ℝ\right)\to \left(A{-}_{ℝ}B=C\leftrightarrow B+C=A\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A{-}_{ℝ}B=C\leftrightarrow B+C=A\right)$$