Theorem mteqand

Description: A modus tollens deduction for inequality. (Contributed by Steven Nguyen, 1-Jun-2023)

	Ref	Expression
Hypotheses	mteqand.1	$$⊢\phi \to C\ne D$$
	mteqand.2	$$⊢\left(\phi \wedge A=B\right)\to C=D$$
Assertion	mteqand	$$⊢\phi \to A\ne B$$

Proof

Step	Hyp	Ref	Expression
1		mteqand.1	$$⊢\phi \to C\ne D$$
2		mteqand.2	$$⊢\left(\phi \wedge A=B\right)\to C=D$$
3	1	neneqd	$$⊢\phi \to \neg C=D$$
4	3 2	mtand	$$⊢\phi \to \neg A=B$$
5	4	neqned	$$⊢\phi \to A\ne B$$