Theorem reladdrsub

Description: Move LHS of a sum into RHS of a (real) difference. Version of mvlladdd with real subtraction. (Contributed by Steven Nguyen, 8-Jan-2023)

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

Proof

Step	Hyp	Ref	Expression
1		reladdrsub.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		reladdrsub.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		reladdrsub.3	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) = 𝐶 )
4	1 2	readdcld	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ∈ ℝ )
5	3 4	eqeltrrd	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
6		resubadd	⊢ ( ( 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐶 −ℝ 𝐴 ) = 𝐵 ↔ ( 𝐴 + 𝐵 ) = 𝐶 ) )
7	3 6	syl5ibrcom	⊢ ( 𝜑 → ( ( 𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐶 −ℝ 𝐴 ) = 𝐵 ) )
8	5 1 2 7	mp3and	⊢ ( 𝜑 → ( 𝐶 −ℝ 𝐴 ) = 𝐵 )
9	8	eqcomd	⊢ ( 𝜑 → 𝐵 = ( 𝐶 −ℝ 𝐴 ) )