Theorem ifnmfalse 51027

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 4490 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)

Assertion

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

Proof of Theorem ifnmfalse

Step	Hyp	Ref	Expression
1		df-nel 3062	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2		iffalse 4490	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → if(𝐴 ∈ 𝐵, 𝐶, 𝐷) = 𝐷)
3	1, 2	sylbi 220	1 ⊢ (𝐴 ∉ 𝐵 → if(𝐴 ∈ 𝐵, 𝐶, 𝐷) = 𝐷)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1569   ∈ wcel 2144   ∉ wnel 3061  ifcif 4481

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nel 3062  df-if 4482

This theorem is used by: (None)