Theorem subadd23

Description: Commutative/associative law for addition and subtraction. (Contributed by NM, 1-Feb-2007)

	Ref	Expression
Assertion	subadd23	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) + 𝐶 ) = ( 𝐴 + ( 𝐶 − 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		addsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) − 𝐵 ) = ( ( 𝐴 − 𝐵 ) + 𝐶 ) )
2		addsubass	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) − 𝐵 ) = ( 𝐴 + ( 𝐶 − 𝐵 ) ) )
3	1 2	eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) + 𝐶 ) = ( 𝐴 + ( 𝐶 − 𝐵 ) ) )
4	3	3com23	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) + 𝐶 ) = ( 𝐴 + ( 𝐶 − 𝐵 ) ) )