Theorem addsub4i

Description: Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 17-Oct-1999)

	Ref	Expression
Hypotheses	negidi.1	⊢ 𝐴 ∈ ℂ
	pncan3i.2	⊢ 𝐵 ∈ ℂ
	subadd.3	⊢ 𝐶 ∈ ℂ
	addsub4i.4	⊢ 𝐷 ∈ ℂ
Assertion	addsub4i	⊢ ( ( 𝐴 + 𝐵 ) − ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 − 𝐶 ) + ( 𝐵 − 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		negidi.1	⊢ 𝐴 ∈ ℂ
2		pncan3i.2	⊢ 𝐵 ∈ ℂ
3		subadd.3	⊢ 𝐶 ∈ ℂ
4		addsub4i.4	⊢ 𝐷 ∈ ℂ
5		addsub4	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) − ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 − 𝐶 ) + ( 𝐵 − 𝐷 ) ) )
6	1 2 3 4 5	mp4an	⊢ ( ( 𝐴 + 𝐵 ) − ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 − 𝐶 ) + ( 𝐵 − 𝐷 ) )