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Theorem ifnmfalse 42371
 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 3948 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 2687 . 2 (𝐴𝐵 ↔ ¬ 𝐴𝐵)
2 iffalse 3948 . 2 𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
31, 2sylbi 205 1 (𝐴𝐵 → if(𝐴𝐵, 𝐶, 𝐷) = 𝐷)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1474   ∈ wcel 1938   ∉ wnel 2685  ifcif 3939 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nel 2687  df-if 3940 This theorem is referenced by: (None)
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