Theorem subsub23d

Description: Swap subtrahend and result of subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	subsub23d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	subsub23d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subsub23d.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	subsub23d	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐴 − 𝐶 ) = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		subsub23d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		subsub23d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subsub23d.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		subsub23	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐴 − 𝐶 ) = 𝐵 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = 𝐶 ↔ ( 𝐴 − 𝐶 ) = 𝐵 ) )