Theorem addneintrd

Description: Introducing a term on the left-hand side of a sum in a negated equality. Contrapositive of addcanad . 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 )
	addneintrd.4	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
Assertion	addneintrd	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ≠ ( 𝐴 + 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		addneintrd.4	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
5	1 2 3	addcand	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) = ( 𝐴 + 𝐶 ) ↔ 𝐵 = 𝐶 ) )
6	5	necon3bid	⊢ ( 𝜑 → ( ( 𝐴 + 𝐵 ) ≠ ( 𝐴 + 𝐶 ) ↔ 𝐵 ≠ 𝐶 ) )
7	4 6	mpbird	⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ≠ ( 𝐴 + 𝐶 ) )