Theorem subcand

Description: Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subaddd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	subcand.4	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐴 − 𝐶 ) )
Assertion	subcand	⊢ ( 𝜑 → 𝐵 = 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		subaddd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		subcand.4	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐴 − 𝐶 ) )
5		subcan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = ( 𝐴 − 𝐶 ) ↔ 𝐵 = 𝐶 ) )
6	1 2 3 5	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = ( 𝐴 − 𝐶 ) ↔ 𝐵 = 𝐶 ) )
7	4 6	mpbid	⊢ ( 𝜑 → 𝐵 = 𝐶 )