Theorem addsubeq4com

Description: Relation between sums and differences. (Contributed by Steven Nguyen, 5-Jan-2023)

	Ref	Expression
Assertion	addsubeq4com	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqcom	⊢ ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐶 + 𝐷 ) = ( 𝐴 + 𝐵 ) )
2		addsubeq4	⊢ ( ( ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ∧ ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ) → ( ( 𝐶 + 𝐷 ) = ( 𝐴 + 𝐵 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐵 ) ) )
3	2	ancoms	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐶 + 𝐷 ) = ( 𝐴 + 𝐵 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐵 ) ) )
4	1 3	syl5bb	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ ( 𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ ) ) → ( ( 𝐴 + 𝐵 ) = ( 𝐶 + 𝐷 ) ↔ ( 𝐴 − 𝐶 ) = ( 𝐷 − 𝐵 ) ) )