Theorem subcanad

Description: Cancellation law for subtraction. Deduction form of subcan . Generalization of subcand . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	negidd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	pncand.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	subaddd.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	subcanad	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = ( 𝐴 − 𝐶 ) ↔ 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		subcan	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 − 𝐵 ) = ( 𝐴 − 𝐶 ) ↔ 𝐵 = 𝐶 ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) = ( 𝐴 − 𝐶 ) ↔ 𝐵 = 𝐶 ) )