Theorem pnpncand

Description: Addition/subtraction cancellation law. (Contributed by Scott Fenton, 14-Dec-2017)

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

Proof

Step	Hyp	Ref	Expression
1		pnpncand.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		pnpncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		pnpncand.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4	2 3	subcld	⊢ ( 𝜑 → ( 𝐵 − 𝐶 ) ∈ ℂ )
5	1 4	addcld	⊢ ( 𝜑 → ( 𝐴 + ( 𝐵 − 𝐶 ) ) ∈ ℂ )
6	5 2 3	subsub2d	⊢ ( 𝜑 → ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) − ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) + ( 𝐶 − 𝐵 ) ) )
7	1 4	pncand	⊢ ( 𝜑 → ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) − ( 𝐵 − 𝐶 ) ) = 𝐴 )
8	6 7	eqtr3d	⊢ ( 𝜑 → ( ( 𝐴 + ( 𝐵 − 𝐶 ) ) + ( 𝐶 − 𝐵 ) ) = 𝐴 )