Theorem ifnefalse

Description: When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs versus applying iffalse directly in this case. It happens, e.g., in oevn0 . (Contributed by David A. Wheeler, 15-May-2015)

	Ref	Expression
Assertion	ifnefalse	$$⊢A\ne B\to if\left(A=B,C,D\right)=D$$

Proof

Step	Hyp	Ref	Expression
1		df-ne	$$⊢A\ne B\leftrightarrow \neg A=B$$
2		iffalse	$$⊢\neg A=B\to if\left(A=B,C,D\right)=D$$
3	1 2	sylbi	$$⊢A\ne B\to if\left(A=B,C,D\right)=D$$