Theorem addcanad

Description: Cancelling a term on the left-hand side of a sum in an equality. Consequence of addcand . (Contributed by David Moews, 28-Feb-2017)

	Ref	Expression
Hypotheses	muld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	addcomd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	addcand.3	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	addcanad.4	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) )
Assertion	addcanad	⊢ ( 𝜑 → 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		addcanad.4	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) )
5	1 2 3	addcand	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) ↔ 𝐵 = 𝐶 ) )
6	4 5	mpbid	⊢ ( 𝜑 → 𝐵 = 𝐶 )