Theorem pnpcan2

Description: Cancellation law for mixed addition and subtraction. (Contributed by Scott Fenton, 9-Jun-2006)

	Ref	Expression
Assertion	pnpcan2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) − ( 𝐵 + 𝐶 ) ) = ( 𝐴 − 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		addcom	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + 𝐶 ) = ( 𝐶 + 𝐴 ) )
2	1	3adant2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + 𝐶 ) = ( 𝐶 + 𝐴 ) )
3		addcom	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 + 𝐶 ) = ( 𝐶 + 𝐵 ) )
4	3	3adant1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐵 + 𝐶 ) = ( 𝐶 + 𝐵 ) )
5	2 4	oveq12d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) − ( 𝐵 + 𝐶 ) ) = ( ( 𝐶 + 𝐴 ) − ( 𝐶 + 𝐵 ) ) )
6		pnpcan	⊢ ( ( 𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐶 + 𝐴 ) − ( 𝐶 + 𝐵 ) ) = ( 𝐴 − 𝐵 ) )
7	6	3coml	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐶 + 𝐴 ) − ( 𝐶 + 𝐵 ) ) = ( 𝐴 − 𝐵 ) )
8	5 7	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) − ( 𝐵 + 𝐶 ) ) = ( 𝐴 − 𝐵 ) )