Theorem addsubeq4d

Description: Relation between sums and differences. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subaddd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	addsub4d.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
Assertion	addsubeq4d	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐶 − 𝐴 ) = ( 𝐵 − 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subaddd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		addsub4d.4	⊢ ( 𝜑 → 𝐷 ∈ ℂ )
5		addsubeq4	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐶 − 𝐴 ) = ( 𝐵 − 𝐷 ) ) )
6	1 2 3 4 5	syl22anc	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐶 − 𝐴 ) = ( 𝐵 − 𝐷 ) ) )