Theorem add42d

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

	Ref	Expression
Hypotheses	addd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	addd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	addd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	add4d.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
Assertion	add42d	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) + ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + ( 𝐷 + 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		addd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		addd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		addd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		add4d.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
5		add42	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) + ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + ( 𝐷 + 𝐵 ) ) )
6	1 2 3 4 5	syl22anc	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) + ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 + 𝐶 ) + ( 𝐷 + 𝐵 ) ) )