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	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	lsubcom23d.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	lsubcom23d.1	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = 𝐶 )
Assertion	lsubcom23d	⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		lsubcom23d.a	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		lsubcom23d.b	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		lsubcom23d.1	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = 𝐶 )
4	1 2	subcld	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) ∈ ℂ )
5	3 4	eqeltrrd	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
6	1 5	subcld	⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) ∈ ℂ )
7	4 3	subeq0bd	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) − 𝐶 ) = 0 )
8	1 5 2	sub32d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐶 ) − 𝐵 ) = ( ( 𝐴 − 𝐵 ) − 𝐶 ) )
9	2	subidd	⊢ ( 𝜑 → ( 𝐵 − 𝐵 ) = 0 )
10	7 8 9	3eqtr4d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐶 ) − 𝐵 ) = ( 𝐵 − 𝐵 ) )
11	6 2 2 10	subcan2d	⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) = 𝐵 )