Theorem addcand

Description: Cancellation law for addition. Theorem I.1 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	muld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	addcomd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	addcand.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	addcand	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) ↔ 𝐵 = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		muld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		addcomd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		addcand.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4		addcan	⊢ ( ( 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 ) )