Theorem subcani

Description: Cancellation law for subtraction. (Contributed by NM, 8-Feb-2005)

	Ref	Expression
Hypotheses	negidi.1	⊢ 𝐴 ∈ ℂ
	pncan3i.2	⊢ 𝐵 ∈ ℂ
	subadd.3	⊢ 𝐶 ∈ ℂ
Assertion	subcani	⊢ ( ( 𝐴 − 𝐵 ) = ( 𝐴 − 𝐶 ) ↔ 𝐵 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		negidi.1	⊢ 𝐴 ∈ ℂ
2		pncan3i.2	⊢ 𝐵 ∈ ℂ
3		subadd.3	⊢ 𝐶 ∈ ℂ
4		subcan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = ( 𝐴 − 𝐶 ) ↔ 𝐵 = 𝐶 ) )
5	1 2 3 4	mp3an	⊢ ( ( 𝐴 − 𝐵 ) = ( 𝐴 − 𝐶 ) ↔ 𝐵 = 𝐶 )