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Theorem ifnefalse 3516
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 versus applying iffalse 3513 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 2328 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 iffalse 3513 . 2 𝐴 = 𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)
31, 2sylbi 120 1 (𝐴𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1335  wne 2327  ifcif 3505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-ne 2328  df-if 3506
This theorem is referenced by:  xnegmnf  9733  rexneg  9734  xaddpnf1  9750  xaddpnf2  9751  xaddmnf1  9752  xaddmnf2  9753  mnfaddpnf  9755  rexadd  9756  fztpval  9985
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