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Metamath Proof Explorer

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 ≠ 𝐵 → if ( 𝐴 = 𝐵 , 𝐶 , 𝐷 ) = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		df-ne	⊢ ( 𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵 )
2		iffalse	⊢ ( ¬ 𝐴 = 𝐵 → if ( 𝐴 = 𝐵 , 𝐶 , 𝐷 ) = 𝐷 )
3	1 2	sylbi	⊢ ( 𝐴 ≠ 𝐵 → if ( 𝐴 = 𝐵 , 𝐶 , 𝐷 ) = 𝐷 )