Theorem ifnmfalse 46351

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 4465 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 3049	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
2		iffalse 4465	. 2 ⊢ (¬ 𝐴 ∈ 𝐵 → if(𝐴 ∈ 𝐵, 𝐶, 𝐷) = 𝐷)
3	1, 2	sylbi 216	1 ⊢ (𝐴 ∉ 𝐵 → if(𝐴 ∈ 𝐵, 𝐶, 𝐷) = 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2108   ∉ wnel 3048  ifcif 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nel 3049  df-if 4457

This theorem is referenced by: (None)