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Metamath Proof Explorer

Theorem addneintr2d

Description: Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad . Consequence of addcan2d . (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 )
	addneintr2d.4	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
Assertion	addneintr2d	⊢ ( 𝜑 → ( 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		addneintr2d.4	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
5	1 2 3	addcan2d	⊢ ( 𝜑 → ( ( 𝐴 + 𝐶 ) = ( 𝐵 + 𝐶 ) ↔ 𝐴 = 𝐵 ) )
6	5	necon3bid	⊢ ( 𝜑 → ( ( 𝐴 + 𝐶 ) ≠ ( 𝐵 + 𝐶 ) ↔ 𝐴 ≠ 𝐵 ) )
7	4 6	mpbird	⊢ ( 𝜑 → ( 𝐴 + 𝐶 ) ≠ ( 𝐵 + 𝐶 ) )