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Metamath Proof Explorer

Theorem add4i

Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999)

Ref Expression
Hypotheses add.1 ⊢ 𝐴 ∈ ℂ
add.2 ⊢ 𝐵 ∈ ℂ
add.3 ⊢ 𝐶 ∈ ℂ
add4.4 ⊢ 𝐷 ∈ ℂ
Assertion add4i ( ( 𝐴 + 𝐵 ) + ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + ( 𝐵 + 𝐷 ) )

Proof

Step Hyp Ref Expression
1 add.1 ⊢ 𝐴 ∈ ℂ
2 add.2 ⊢ 𝐵 ∈ ℂ
3 add.3 ⊢ 𝐶 ∈ ℂ
4 add4.4 ⊢ 𝐷 ∈ ℂ
5 add4 ⊢ ( ( ( 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 ) )