Theorem lsubcom23d

Description: Swap the second and third variables in an equation with subtraction on the left, converting it into an addition. (Contributed by SN, 23-Aug-2024)

	Ref	Expression
Hypotheses	lsubcom23d.a	$$⊢\phi \to A\in ℂ$$
	lsubcom23d.b	$$⊢\phi \to B\in ℂ$$
	lsubcom23d.1	$$⊢\phi \to A-B=C$$
Assertion	lsubcom23d	$$⊢\phi \to A-C=B$$

Proof

Step	Hyp	Ref	Expression
1		lsubcom23d.a	$$⊢\phi \to A\in ℂ$$
2		lsubcom23d.b	$$⊢\phi \to B\in ℂ$$
3		lsubcom23d.1	$$⊢\phi \to A-B=C$$
4	1 2	subcld	$$⊢\phi \to A-B\in ℂ$$
5	3 4	eqeltrrd	$$⊢\phi \to C\in ℂ$$
6	1 5	subcld	$$⊢\phi \to A-C\in ℂ$$
7	4 3	subeq0bd	$$⊢\phi \to A-B-C=0$$
8	1 5 2	sub32d	$$⊢\phi \to A-C-B=A-B-C$$
9	2	subidd	$$⊢\phi \to B-B=0$$
10	7 8 9	3eqtr4d	$$⊢\phi \to A-C-B=B-B$$
11	6 2 2 10	subcan2d	$$⊢\phi \to A-C=B$$