Theorem lsubswap23d

Description: Swap the second and third variables in an equation with subtraction on the left, converting it into an addition.

EDITORIAL: The label for this theorem is questionable. Do not move until it would have 7 uses: current additional uses: (none). (Contributed by SN, 23-Aug-2024)

	Ref	Expression
Hypotheses	lsubswap23d.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	lsubswap23d.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	lsubswap23d.1	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = 𝐶 )
Assertion	lsubswap23d	⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		lsubswap23d.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		lsubswap23d.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		lsubswap23d.1	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = 𝐶 )
4	1 2	subcld	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) ∈ ℂ )
5	3 4	eqeltrrd	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
6	1 2 3	lsubrotld	⊢ ( 𝜑 → ( 𝐵 + 𝐶 ) = 𝐴 )
7	6	eqcomd	⊢ ( 𝜑 → 𝐴 = ( 𝐵 + 𝐶 ) )
8	2 5 7	mvrraddd	⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) = 𝐵 )