 Mathbox for David A. Wheeler < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ifnmfalse Structured version   Visualization version   GIF version

Theorem ifnmfalse 43031
 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 4234 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 3047 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 iffalse 4234 . 2 𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
31, 2sylbi 207 1 (𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1631   ∈ wcel 2145   ∉ wnel 3046  ifcif 4225 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nel 3047  df-if 4226 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator