Theorem resubaddd

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

	Ref	Expression
Hypotheses	resubaddd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	resubaddd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	resubaddd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
Assertion	resubaddd	⊢ ( 𝜑 → ( ( 𝐴 −ℝ 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		resubaddd.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		resubaddd.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		resubaddd.3	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4		resubadd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 −ℝ 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 −ℝ 𝐵 ) = 𝐶 ↔ ( 𝐵 + 𝐶 ) = 𝐴 ) )