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Theorem ifnefalse 3537
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 3534 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 2341 . 2 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
2 iffalse 3534 . 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 1348  wne 2340  ifcif 3526
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 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ne 2341  df-if 3527
This theorem is referenced by:  xnegmnf  9786  rexneg  9787  xaddpnf1  9803  xaddpnf2  9804  xaddmnf1  9805  xaddmnf2  9806  mnfaddpnf  9808  rexadd  9809  fztpval  10039  pcval  12250  lgsval3  13713  lgsdinn0  13743
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