Theorem add42

Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 12-May-2005)

	Ref	Expression
Assertion	add42	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) + ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + ( 𝐷 + 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		add4	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) + ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + ( 𝐵 + 𝐷 ) ) )
2		addcom	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐷 ∈ ℂ ) → ( 𝐵 + 𝐷 ) = ( 𝐷 + 𝐵 ) )
3	2	ad2ant2l	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( 𝐵 + 𝐷 ) = ( 𝐷 + 𝐵 ) )
4	3	oveq2d	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐶 ) + ( 𝐵 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + ( 𝐷 + 𝐵 ) ) )
5	1 4	eqtrd	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) + ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + ( 𝐷 + 𝐵 ) ) )