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Theorem ifnefalse 3670
 Description: When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3669 directly in this case. It happens, e.g., in oevn0 in set.mm. (Contributed by David A. Wheeler, 15-May-2015.)
Assertion
Ref Expression
ifnefalse (AB → if(A = B, C, D) = D)

Proof of Theorem ifnefalse
StepHypRef Expression
1 df-ne 2518 . 2 (AB ↔ ¬ A = B)
2 iffalse 3669 . 2 A = B → if(A = B, C, D) = D)
31, 2sylbi 187 1 (AB → if(A = B, C, D) = D)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ≠ wne 2516   ifcif 3662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-ne 2518  df-if 3663 This theorem is referenced by: (None)
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