Metamath Proof Explorer


Theorem subadd4d

Description: Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses negidd.1 ( 𝜑𝐴 ∈ ℂ )
pncand.2 ( 𝜑𝐵 ∈ ℂ )
subaddd.3 ( 𝜑𝐶 ∈ ℂ )
addsub4d.4 ( 𝜑𝐷 ∈ ℂ )
Assertion subadd4d ( 𝜑 → ( ( 𝐴𝐵 ) − ( 𝐶𝐷 ) ) = ( ( 𝐴 + 𝐷 ) − ( 𝐵 + 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 negidd.1 ( 𝜑𝐴 ∈ ℂ )
2 pncand.2 ( 𝜑𝐵 ∈ ℂ )
3 subaddd.3 ( 𝜑𝐶 ∈ ℂ )
4 addsub4d.4 ( 𝜑𝐷 ∈ ℂ )
5 subadd4 ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴𝐵 ) − ( 𝐶𝐷 ) ) = ( ( 𝐴 + 𝐷 ) − ( 𝐵 + 𝐶 ) ) )
6 1 2 3 4 5 syl22anc ( 𝜑 → ( ( 𝐴𝐵 ) − ( 𝐶𝐷 ) ) = ( ( 𝐴 + 𝐷 ) − ( 𝐵 + 𝐶 ) ) )