Theorem ifnmfalse

Description: If A is not a member of B, but an "if" condition requires it, 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. (Contributed by David A. Wheeler, 15-May-2015)

	Ref	Expression
Assertion	ifnmfalse	⊢ ( 𝐴 ∉ 𝐵 → if ( 𝐴 ∈ 𝐵 , 𝐶 , 𝐷 ) = 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		df-nel	⊢ ( 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 )