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Metamath Proof Explorer

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 ) )