Theorem pnpcand

Description: Cancellation law for mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subaddd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	pnpcand	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) − ( 𝐴 + 𝐶 ) ) = ( 𝐵 − 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subaddd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		pnpcan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐵 ) − ( 𝐴 + 𝐶 ) ) = ( 𝐵 − 𝐶 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) − ( 𝐴 + 𝐶 ) ) = ( 𝐵 − 𝐶 ) )