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Theorem ifnmfalse 42303
 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 vs. applying iffalse 4045 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 2783 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 iffalse 4045 . 2 𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
31, 2sylbi 206 1 (𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ifcif 4036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nel 2783  df-if 4037 This theorem is referenced by: (None)
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