Theorem subsub

Description: Law for double subtraction. (Contributed by NM, 13-May-2004)

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

Proof

Step	Hyp	Ref	Expression
1		subsub2	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( 𝐵 − 𝐶 ) ) = ( 𝐴 + ( 𝐶 − 𝐵 ) ) )
2		addsubass	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) − 𝐵 ) = ( 𝐴 + ( 𝐶 − 𝐵 ) ) )
3		addsub	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) − 𝐵 ) = ( ( 𝐴 − 𝐵 ) + 𝐶 ) )
4	2 3	eqtr3d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + ( 𝐶 − 𝐵 ) ) = ( ( 𝐴 − 𝐵 ) + 𝐶 ) )
5	4	3com23	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 + ( 𝐶 − 𝐵 ) ) = ( ( 𝐴 − 𝐵 ) + 𝐶 ) )
6	1 5	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 − 𝐵 ) + 𝐶 ) )