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Theorem ifnmfalse 45786
 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 4432 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
Assertion
Ref Expression
ifnmfalse (𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)

Proof of Theorem ifnmfalse
StepHypRef Expression
1 df-nel 3056 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 iffalse 4432 . 2 𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
31, 2sylbi 220 1 (𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2111   ∉ wnel 3055  ifcif 4423 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nel 3056  df-if 4424 This theorem is referenced by: (None)
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