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Theorem ifnmfalse 47294
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 4496 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 4496 . 2 𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
31, 2sylbi 216 1 (𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  wnel 3046  ifcif 4487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nel 3047  df-if 4488
This theorem is referenced by: (None)
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