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Theorem ifnmfalse 44227
 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 4359 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 3075 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 iffalse 4359 . 2 𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
31, 2sylbi 209 1 (𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1507   ∈ wcel 2050   ∉ wnel 3074  ifcif 4350 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-ext 2751 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-ex 1743  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nel 3075  df-if 4351 This theorem is referenced by: (None)
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