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Theorem ifnefalse 3342
 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 3339 directly in this case. (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 2206 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 iffalse 3339 . 2 𝐴 = 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)
31, 2sylbi 114 1 (𝐴𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1243   ≠ wne 2204  ifcif 3331 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-if 3332 This theorem is referenced by:  xnegmnf  8742  rexneg  8743  fztpval  8945
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