Theorem subsub3

Description: Law for double subtraction. (Contributed by NM, 27-Jul-2005)

	Ref	Expression
Assertion	subsub3	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 + 𝐶 ) − 𝐵 ) )

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	2	3com23	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 + 𝐶 ) − 𝐵 ) = ( 𝐴 + ( 𝐶 − 𝐵 ) ) )
4	1 3	eqtr4d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 − ( 𝐵 − 𝐶 ) ) = ( ( 𝐴 + 𝐶 ) − 𝐵 ) )